*Article* **The Riemann Zeros as Spectrum and the Riemann Hypothesis**

## **Germán Sierra**

Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain; german.sierra@uam.es

Received: 31 December 2018; Accepted: 26 March 2019; Published: 4 April 2019

**Abstract:** We present a spectral realization of the Riemann zeros based on the propagation of a massless Dirac fermion in a region of Rindler spacetime and under the action of delta function potentials localized on the square free integers. The corresponding Hamiltonian admits a self-adjoint extension that is tuned to the phase of the zeta function, on the critical line, in order to obtain the Riemann zeros as bound states. The model suggests a proof of the Riemann hypothesis in the limit where the potentials vanish. Finally, we propose an interferometer that may yield an experimental observation of the Riemann zeros.

**Keywords:** zeta function; Pólya-Hilbert conjecture; Riemann interferometer

## **1. Introduction**

One of the most promising approaches to prove the Riemann Hypothesis [1–7] is based on the conjecture, due to Pólya and Hilbert, that the Riemann zeros are the eigenvalues of a quantum mechanical Hamiltonian [8]. This bold idea is supported by several results and analogies involving Number Theory, Random Matrix Theory and Quantum Chaos [9–17]. However, the construction of a Hamiltonian whose spectrum contains the Riemann zeros, has eluded researchers for several decades. In this paper we shall review the progress made along this direction starting from the famous *xp* model proposed in 1999 by Berry, Keating and Connes [18–20] that inspired many works [21–45], some of them will be discuss below. See [46] for a general review on physical approaches to the RH. Other approaches to the RH and related material can be found in [47–63].

To relate *xp* with the Riemann zeros, Berry, Keating and Connes used two different regularizations. The Berry and Keating regularization led to a discrete spectrum related to the smooth Riemann zeros [18,19], while Connes's regularization led to an absorption spectrum where the *zeros* are missing spectral lines [20]. A physical realization of the Connes model was obtained in 2008 in terms of the dynamics of an electron moving in two dimensions under the action of a uniform perpendicular magnetic field and an electrostatic potential [29]. However this model has not been able to reproduce the exact location of the Riemann zeros. On the other hand, the Berry–Keating *xp* model was revisited in 2011 in terms of the classical Hamiltonians *H* = *x*(*p* + 1/*p*), and *H* = (*x* + 1/*x*)(*p* + 1/*p*) whose quantizations contain the smooth approximation of the Riemann zeros [32,36]. Later on, these models were generalized in terms of the family of Hamiltonians *H* = *U*(*x*)*p* + *V*(*x*)/*p* that were shown to describe the dynamics of a massive particle in a relativistic spacetime whose metric can be constructed using the functions *U* and *V* [35]. This result suggested a reformulation of *H* = *U*(*x*)*p* + *V*(*x*)/*p* in terms of the massive Dirac equation in the aforementioned spacetimes [38]. Using this reformulation, the Hamiltonian *H* = *x*(*p* + 1/*p*) was shown to be equivalent to the massive Dirac equation in Rindler spacetime that is the natural arena to study accelerated observers and the Unruh effect [42]. This result provides an appealing spacetime interpretation of the *xp* model and in particular of the smooth Riemann zeros.

To obtain the exact *zeros*, one must make further modifications of the Dirac model. First, the fermion must become massless. This change is suggested by a field theory interpretation of the Pólya's *ξ* function and its comparison with the Riemann's *ξ* function. On the other hand, inspired by the Berry's conjecture on the relation between prime numbers and periodic orbits [12,14] we incorporated the prime numbers into the Dirac action by means of Dirac delta functions [42]. These delta functions represent moving mirrors that reflect or transmit massless fermions. The spectrum of the complete model can be analyzed using transfer matrix techniques that can be solved exactly in the limit where the reflection amplitudes of the mirrors go to zero that is when the mirrors become transparent. In this limit we find that the *zeros* on the critical line are eigenvalues of the Hamiltonian by choosing appropriately the parameter that characterizes the self-adjoint extension of the Hamiltonian. One obtains in this manner a spectral realization of the Riemann zeros that differs from the Pólya and Hilbert conjecture in the sense that one needs to fine tune a parameter to *see* each individual *zero*. In our approach we are not able to find a single Hamiltonian encompassing all the *zeros* at once. Finally, we propose an experimental realization of the Riemann zeros using an interferometer consisting of an array of semitransparent mirrors, or beam splitters, placed at positions related to the logarithms of the square free integers.

The paper is organized in a historical and pedagogical way presenting at the end of each section a summary of achievements ( ), shortcomings/obstacles (✕) and questions/suggestions (?).

## **2. The Semiclassical** *XP* **Berry, Keating and Connes Model**

In this section, we review the main results concerning the classical and semiclassical *xp* model [18–20]. A classical trajectory of the Hamiltonian *H* = *xp*, with energy *E*, is given by

$$\mathbf{x}(t) = \mathbf{x}\_0 \,\mathrm{e}^t, \qquad \mathbf{p}(t) = p\_0 \,\mathrm{e}^{-t}, \qquad \mathrm{E} = \mathbf{x}\_0 p\_0 \,\mathrm{e} \tag{1}$$

that traces the parabola *E* = *xp* in phase space plotted in Figure 1. *E* has the dimension of an action, so one should multiply *xp* by a frequency to get an energy, but for the time being we keep the notation *H* = *xp*. Under a time reversal transformation, *x* → *x*, *p* → −*p* one finds *xp* → −*xp*, so that this symmetry is broken. This is why reversing the time variable *t* in (1) does not yield a trajectory generated by *xp*. As *t* → ∞, the trajectory becomes unbounded that is |*x*| → ∞, so one expects the semiclassical and quantum spectrum of the *xp* model to form a continuum. To get a discrete spectrum Berry and Keating introduced the constraints |*x*| ≥ *<sup>x</sup>* and |*p*| ≥ *<sup>p</sup>*, so that the particle starts at *t* = 0 at (*x*, *p*)=(*<sup>x</sup>*, *E*/*<sup>x</sup>*) and ends at (*x*, *p*)=(*E*/*<sup>p</sup>*, *<sup>p</sup>*) after a time lapse *T* = log(*E*/*x<sup>p</sup>*) (we assume for simplicity that *x*, *p* > 0). The trajectories are now bounded, but not periodic. A semiclassical estimate of the number of energy levels, *n*BK(*E*), between 0 and *E* > 0 is given by the formula

$$m\_{\rm BK}(E) = \frac{A\_{\rm BK}}{2\pi\hbar} = \frac{E}{2\pi\hbar} \left( \log \frac{E}{\ell\_x \ell\_p} - 1 \right) + \frac{7}{8'} \tag{2}$$

where *A*BK is the phase space area below the parabola *E* = *xp* and the lines *x* = *<sup>x</sup>* and *p* = *p*, measured in units of the Planck's constant 2*πh*¯ (see Figure 1). The term 7/8 arises from the Maslow phase [18]. In the course of the paper, we shall encounter this equation several times with the constant term depending on the particular model.

**Figure 1.** (**Left**): The region in shadow describes the allowed phase space with area *A*BK bounded by the classical trajectory (1) with *E* > 0 and the constraints *x* ≥ *<sup>x</sup>*, *p* ≥ *p*. (**Right**): Same as before with the constraints 0 < *x*, *p* < Λ.

Berry and Keating compared this result with the average number of Riemann zeros, whose imaginary part is less than *t* with *t* 1,

$$
\langle n(t)\rangle \simeq \frac{t}{2\pi} \left(\log\frac{t}{2\pi} - 1\right) + \frac{7}{8} + O(1/t) \,\tag{3}
$$

finding an agreement with the identifications

$$
\hbar t = \frac{E}{\hbar'} \qquad \ell\_x \,\ell\_p = 2\pi\hbar. \tag{4}
$$

Thus, the semiclassical energies *E*, expressed in units of *h*¯, are identified with the Riemann zeros, while *xp* is identified with the Planck's constant. This result is remarkable given the simplicity of the assumptions. However, one must observe that the derivation of Equation (2) is heuristic, so one goal is to find a consistent quantum version of it.

Connes proposed another regularization of the *xp* model based on the restrictions |*x*| ≤ Λ and |*p*| ≤ Λ, where Λ is a common cutoff, which is taken to infinity at the end of the calculation [20]. The semiclassical number of states is computed as before yielding (see Figure 1, we set ¯*h* = 1)

$$m\_{\mathbb{C}}(E) = \frac{A\_{\mathbb{C}}}{2\pi} = \frac{E}{2\pi} \log \frac{\Lambda^2}{2\pi} - \frac{E}{2\pi} \left( \log \frac{E}{2\pi} - 1 \right) \,. \tag{5}$$

The first term on the RHS of this formula diverges in the limit Λ → ∞, which corresponds to a continuum of states. The second term is minus the average number of Riemann zeros, which according to Connes, become missing spectral lines in the continuum [17,20]. This is called the *absorption* spectral interpretation of the Riemann zeros, as opposed to the standard *emission* spectral interpretation where the *zeros* form a discrete spectrum. Connes, relates the minus sign in Equation (5) to a minus sign discrepancy between the fluctuation term of the number of zeros and the associated formula in the theory of Quantum Chaos. We shall show below that the negative term in Equation (5) must be seen as a finite size correction of discrete energy levels and not as an indication of missing spectral lines.

Let us give for completeness the formula for the exact number of *zeros* up to *t* [2,3]

$$\begin{aligned} n\_{\mathbb{R}}(t) &= \langle n(t) \rangle + n\_{\mathbb{R}}(t), \\ \langle n(t) \rangle &= \frac{\theta(t)}{\pi} + 1, \qquad n\_{\mathbb{R}}(t) = \frac{1}{\pi} \text{Im } \log \zeta \left( \frac{1}{2} + it \right), \end{aligned} \tag{6}$$

where *n*(*t*) is the Riemann–von Mangoldt formula that gives the average behavior in terms of the function *θ*(*t*)

$$\theta(t) \quad = \text{Im}\, \log \Gamma \left(\frac{1}{4} + \frac{it}{2}\right) - \frac{t}{2} \log \pi \stackrel{t \to \infty}{\longrightarrow} \frac{t}{2} \log \frac{t}{2\pi} - \frac{t}{2} - \frac{\pi}{8} + O(1/t) \tag{7}$$

that can also be written as

$$
\epsilon^{2i\theta(t)} = \pi^{-it} \frac{\Gamma\left(\frac{1}{4} + \frac{it}{2}\right)}{\Gamma\left(\frac{1}{4} - \frac{it}{2}\right)}.\tag{8}
$$

*θ*(*t*) is the phase of the Riemann zeta function on the critical line, that can be expressed as

$$
\zeta\left(\frac{1}{2} + it\right) = e^{-i\theta(t)}\ Z(t),
\tag{9}
$$

where *Z*(*t*) is the Riemann-Siegel zeta function, or Hardy function, that on the critical line satisfies

$$Z(t) = Z(-t) = Z^\*(t), \qquad t \in \mathbb{R}.\tag{10}$$

**Summary:**

The semiclassical spectrum of the *xp* Hamiltonian reproduces the average Riemann zeros.


## **3. The Quantum** *XP* **Model**

To quantize the *xp* Hamiltonian, Berry and Keating used the normal ordered operator [18]

$$\hat{H} = \frac{1}{2}(\mathbf{x}\,\hat{p} + \hat{p}\,\mathbf{x}) = -i\hbar\left(\mathbf{x}\frac{d}{dx} + \frac{1}{2}\right), \qquad \mathbf{x} \in \mathbb{R}, \tag{11}$$

where *x* belongs to the real line and *p*ˆ = −*ihd*¯ /*dx* is the momentum operator. We shall show below that despite of being a natural quantization of the classical *xp* Hamiltonian, it does not reproduce the semiclassical spectrum obtained in the previous section. It is, however, of great interest to study it in detail since it is the basis of the rest of the work.

It is convenient to restrict *x* to the positive half-line, then (11) is equivalent to the expression

$$
\hat{H} = \sqrt{\mathbf{x}} \,\not\!\!\!\!\!\/ \sqrt{\mathbf{x}}, \qquad \mathbf{x} \ge \mathbf{0}. \tag{12}
$$

*H*ˆ is an essentially self-adjoint operator acting on the Hilbert space *L*2(0, ∞) of square integrable functions in the half-line R+ = (0, ∞) [23,24,30]. The eigenfunctions, with eigenvalue *E*, are given by

$$\psi\_E(\mathbf{x}) = \frac{1}{\sqrt{2\pi\hbar}} \mathbf{x}^{-\frac{1}{2} + \frac{iE}{\hbar}}, \qquad \mathbf{x} > \mathbf{0}, \quad E \in \mathbb{R} \tag{13}$$

and the spectrum is the real line R. The normalization of (13) is given by the Dirac's delta function

$$
\langle \Psi\_{\rm E} | \Psi\_{\rm E'} \rangle = \int\_0^\infty dx \, \Psi\_{\rm E}^\*(x) \, \Psi\_{\rm E'}(x) = \delta(E - E'). \tag{14}
$$

The eigenfunctions (13) form an orthonormal basis of *L*2(0, ∞), that is related to the Mellin transform in the same manner that the eigenfunctions of the momentum operator *p*ˆ, on the real line, are related to the Fourier transform [24]. If one takes *x* in the whole real line, then the spectrum of the Hamiltonian (11) is doubly degenerate. This degeneracy can be understood from the invariance of *xp*

under the parity transformation *x* → −*x*, *p* → −*p*, which allows one to split the eigenfunctions with energy *E* into even and odd sectors

$$\psi\_{\rm E}^{(e)}(\mathbf{x}) = \frac{1}{\sqrt{2\pi\hbar}} |\mathbf{x}|^{-\frac{1}{2} + \frac{i\overline{\mathbf{r}}}{\hbar}}, \quad \psi\_{\rm E}^{(o)}(\mathbf{x}) = \frac{\text{sign}\,\mathbf{x}}{\sqrt{2\pi\hbar}} |\mathbf{x}|^{-\frac{1}{2} + \frac{i\overline{\mathbf{r}}}{\hbar}}, \qquad \mathbf{x} \in \mathbb{R}, \quad E \in \mathbb{R}. \tag{15}$$

Berry and Keating computed the Fourier transform of the even wave function *ψ*(*e*) *<sup>E</sup>* (*x*) [18]

$$\psi\_{E}^{(\epsilon)}(p) = \frac{1}{\sqrt{2\pi\hbar}} \int\_{-\infty}^{\infty} dx \,\psi\_{E}^{(\epsilon)}(x) \, e^{-ipx/\hbar} \tag{16}$$

$$= \frac{1}{\sqrt{2\pi\hbar}} |p|^{-\frac{1}{2} - \frac{i\mathbb{E}}{\hbar}} (2\hbar)^{iE/\hbar} \frac{\Gamma\left(\frac{1}{4} + \frac{iE}{2\hbar}\right)}{\Gamma\left(\frac{1}{4} - \frac{iE}{2\hbar}\right)},$$

which means that the position and momentum eigenfunctions are each other's time reversed, giving a physical interpretation of the phase *θ*(*t*), see Equation (8). Choosing odd eigenfunctions leads to an equation similar to Equation (16) in terms of the gamma functions Γ( <sup>3</sup> <sup>4</sup> <sup>±</sup> *iE* <sup>2</sup> ) that appear in the functional relation of the odd Dirichlet *L*-functions. Equation (16) is a consequence of the exchange *x* ↔ *p* symmetry of the *xp* Hamiltonian, which is an important ingredient of the *xp* model.

#### **Comments:**


$$\psi\_{\mathbb{E}}(\mathbf{x}) \to \sum\_{n=1}^{\infty} \psi\_{\mathbb{E}}(n\mathbf{x}) = \frac{1}{\sqrt{2\pi\hbar}} \mathbf{x}^{-\frac{1}{2} + \frac{i\mathbb{E}}{\hbar}} \sum\_{n=1}^{\infty} \frac{1}{n^{\frac{1}{2} - \frac{i\mathbb{E}}{\hbar}}} = \frac{1}{\sqrt{2\pi\hbar}} \mathbf{x}^{-\frac{1}{2} + \frac{i\mathbb{E}}{\hbar}} \zeta(1/2 - i\mathbb{E}/\hbar) \,. \tag{17}$$

If there exists a physical reason for this quantity to vanish one would obtain the Riemann zeros *En*. Equation (17) could be interpreted as the breaking of the continuous scale invariance to discrete scale invariance.

#### **Summary:**

✕ The normal order quantization of *xp* does not exhibit any trace of the Riemann zeros. The phase of the zeta function appears in the Fourier transform of the *xp* eigenfunctions.

## **4. The Landau Model and** *XP*

Let us consider a charged particle moving in a plane under the action of a perpendicular magnetic field and an electrostatic potential *V*(*x*, *y*) ∝ *xy* [29]. The Langrangian describing the dynamics is given, in the Landau gauge, by

$$\mathcal{L} = \frac{\mu}{2} (\dot{\mathbf{x}}^2 + \dot{\mathbf{y}}^2) - \frac{eB}{c} \dot{\mathbf{y}} \mathbf{x} - e\lambda x \mathbf{y} \, \, \, \, \tag{18}$$

where *μ* is the mass, *e* the electric charge, *B* the magnetic field, *c* the speed of light and *λ* a coupling constant that parameterizes the electrostatic potential. There are two normal modes with real, *ωc*, and imaginary, *ωh*, angular frequencies, describing a cyclotronic and a hyperbolic motion respectively. In the limit where *ω<sup>c</sup>* >> |*ωh*|, only the Lowest Landau Level (LLL) is relevant and the effective Lagrangian becomes

$$\mathcal{L}\_{\rm eff} = p\dot{\mathbf{x}} - |\omega\_h| \mathbf{x} p\_{\prime} \qquad p = \frac{\hbar y}{\ell^2} , \qquad \ell = \left(\frac{\hbar c}{eB}\right)^{1/2} , \tag{19}$$

where is the magnetic length, which is proportional to the radius of the cyclotronic orbits in the LLL. The coordinates *x* and *y*, which commute in the 2D model, after the projection to the LLL, become canonical conjugate variables, and the effective Hamiltonian is proportional to the *xp* Hamiltonian with the proportionality constant given by the angular frequency |*ωh*| (this is the missing frequency factor mentioned in Section 2). The quantum Hamiltonian associated with the Lagrangian (18) is

$$
\hat{H} = \frac{1}{2\mu} \left[ \mathfrak{p}\_{\text{x}} + \left( \mathfrak{p}\_{\text{y}} + \frac{\hbar}{\ell^2} \mathbf{x} \right)^2 \right] + e\lambda \mathbf{x} \mathbf{y} \,, \tag{20}
$$

where *p*ˆ*<sup>x</sup>* = −*ih*¯ *∂<sup>x</sup>* and *p*ˆ*<sup>y</sup>* = −*ih*¯ *∂y*. After a unitary transformation (20) becomes the sum of two commuting Hamiltonians corresponding to the cyclotronic and hyperbolic motions alluded to above

$$\begin{aligned} H &=& H\_{\mathfrak{c}} + H\_{\mathfrak{h}}, \\ H\_{\mathfrak{c}} &=& \frac{\omega\_{\mathfrak{c}}}{2} (\not{p}^2 + q^2), \end{aligned} \tag{21}$$

$$H\_{\mathfrak{h}} = \begin{aligned} \frac{\omega\_{\mathfrak{c}}}{2} (\not{p}^2 + q^2), \end{aligned} \qquad H\_{\mathfrak{h}} = \frac{|\omega\_{\mathfrak{h}}|}{2} (\not{p} Q + Q \not{p}).$$

In the limit *ω<sup>c</sup>* |*ωh*| one has

$$
\omega\_{\rm t} \simeq \frac{eB}{\mu c'} \quad |\omega\_{\rm t}| \sim \frac{\lambda c}{B} \,. \tag{22}
$$

The unitary transformation that brings Equation (20) into Equation (21) corresponds to the classical canonical transformation

$$q = x + p\_{y}, \quad p = p\_{x}, \quad Q = -p\_{y}, \quad P = y + p\_{x} \,. \tag{23}$$

When *ω<sup>c</sup>* |*ωh*|, the low energy states of *H* are the product of the lowest eigenstate of *Hc*, namely *ψ* = *e*−*q*2/2-2 , times the eigenstates of *Hh* that can be chosen as even or odd under the parity transformation *Q* → −*Q*

$$\Phi\_E^+(Q) = \frac{1}{|Q|^{\frac{1}{2} - iE'}}, \qquad \Phi\_E^-(Q) = \frac{\text{sign}(Q)}{|Q|^{\frac{1}{2} - iE}}.\tag{24}$$

The corresponding wave functions are given by (we choose |*ωh*| = 1)

$$\psi\_{\underline{E}}^{\pm}(\mathbf{x},\mathbf{y}) = \mathbb{C} \int d\mathbf{Q} \, e^{-i\mathbf{Q}y/\ell^{2}} e^{-(\mathbf{x}-\mathbf{Q})^{2}/2\ell^{2}} \Phi\_{\underline{E}}^{\pm}(\mathbf{Q}) \, , \tag{25}$$

where *C* is a normalization constant, which yields

$$\begin{split} \psi\_{E}^{+}(\mathbf{x},\mathbf{y}) &= \quad \mathbb{C}\_{E}^{+} e^{-\frac{\mathbf{x}^{2}}{2\ell^{2}}} M\left(\frac{1}{4} + \frac{iE}{2}, \frac{1}{2}, \frac{(\mathbf{x} - i\mathbf{y})^{2}}{2\ell^{2}}\right), \\ \psi\_{E}^{-}(\mathbf{x},\mathbf{y}) &= \quad \mathbb{C}\_{E}^{-}(\mathbf{x} - i\mathbf{y}) e^{-\frac{\mathbf{x}^{2}}{2\ell^{2}}} M\left(\frac{3}{4} + \frac{iE}{2}, \frac{3}{2}, \frac{(\mathbf{x} - i\mathbf{y})^{2}}{2\ell^{2}}\right), \end{split} \tag{26}$$

where *M*(*a*, *b*, *z*) is a confluent hypergeometric function [64]. Figure 2 shows that the maximum of the absolute value of *ψ*<sup>+</sup> *<sup>E</sup>* is attained on the classical trajectory *E* = *xy* (in units of *h*¯ = - = 1). This 2D representation of the classical trajectories is possible because in the LLL *x* and *y* become canonical conjugate variables and consequently the 2D plane coincides with the phase space (*x*, *p*).

To count the number of states with an energy below *E* one places the particle into a box: |*x*| < *L*, |*y*| < *L* and impose the boundary conditions

$$
\psi\_E^+ (\mathbf{x}, L) = \epsilon^{i \ge L/\ell^2} \psi\_E^+ (L, \mathbf{x}) \,, \tag{27}
$$

which identifies the outgoing particle at *x* = *L* with the incoming particle at *y* = *L* up to a phase. The asymptotic behavior *L* of (26) is

$$\psi\_{E}^{+}(L,\mathbf{x}) \quad \simeq \quad e^{-i\mathbf{x}L/\ell^{2} - \mathbf{x}^{2}/2\ell^{2}} \frac{\Gamma\left(\frac{1}{2}\right)}{\Gamma\left(\frac{1}{4} + \frac{i\overline{E}}{2}\right)} \left(\frac{L^{2}}{2\ell^{2}}\right)^{-\frac{1}{4} + \frac{i\overline{E}}{2}},\tag{28}$$
 
$$\psi\_{E}^{+}(\mathbf{x},L) \quad \simeq \quad e^{-\mathbf{x}^{2}/2\ell^{2}} \frac{\Gamma\left(\frac{1}{2}\right)}{\Gamma\left(\frac{1}{4} - \frac{i\overline{E}}{2}\right)} \left(\frac{L^{2}}{2\ell^{2}}\right)^{-\frac{1}{4} - \frac{i\overline{E}}{2}},$$

that plugged into the BC (27) yields

$$\frac{\Gamma\left(\frac{1}{4} + \frac{iE}{2}\right)}{\Gamma\left(\frac{1}{4} - \frac{iE}{2}\right)} \left(\frac{L^2}{2\ell^2}\right)^{-iE} = 1,\tag{29}$$

or using Equation (8)

$$\epsilon^{2i\theta(\mathbb{E})} \left(\frac{L^2}{2\pi\ell^2}\right)^{-i\mathbb{E}} = 1. \tag{30}$$

Hence the number of states *n*(*E*) with energy less that *E* is given by

$$n(E) \simeq \frac{E}{2\pi} \log\left(\frac{L^2}{2\pi\ell^2}\right) + 1 - \left< n(E) \right>\,\tag{31}$$

whose asymptotic behavior coincides with Connes's Formula (5) for a cutoff Λ = *L*/-. In fact, the term *n*(*E*) is the exact Riemann–von Mangoldt Formula (6).

**Figure 2.** Plot of <sup>|</sup>*ψ*<sup>+</sup> *<sup>E</sup>* (*x*, *y*)| for *E* = 10 in the region −10 < *x*, *y* < 10. **Left**: 3D representation, **Right**: density plot.

#### **Summary:**

The Landau model with a *xy* potential provides a physical realization of Connes's *xp* model. The finite size effects in the spectrum are given by the Riemann–von Mangoldt formula. ✕ There are no missing spectral lines in the physical realizations of *xp* à la Connes.

## **5. The** *XP* **Model Revisited**

An intuitive argument of why the quantum Hamiltonian (*xp*ˆ + *px*ˆ )/2 has a continuum spectrum is that the classical trajectories of *xp* are unbounded. Therefore, to have a discrete spectrum one should modify *xp* to bound the trajectories. This is achieved by the classical Hamiltonian [32]

$$H\_I = \mathbf{x}\left(p + \frac{\ell\_p^2}{p}\right), \quad \mathbf{x} \ge \ell\_\mathbf{x}. \tag{32}$$

For |*p*| >> *<sup>p</sup>*, a classical trajectory with energy *E* satisfies *E xp*, but for |*p*| ∼ *p*, the coordinate |*x*| slows down, reaches a maximum and goes back to the value *x*, where it bounces off starting again at high momentum. In this manner one gets a periodic orbit (see Figure 3)

**Figure 3.** Classical trajectories of the Hamiltonians (32) (**left**) and (35) (**right**) in phase space with *E* > 0. The dashed lines denote the hyperbola *E* = *xp*. (*<sup>x</sup>*, *<sup>p</sup>*) is a fixed-point solution of the classical equations generated by (32) and (35).

$$\begin{array}{rcl} x(t) &=& \frac{\ell\_x}{|p\_0|} e^{2t} \sqrt{(p\_0^2 + \ell\_p^2)e^{-2t} - \ell\_{p'}^2} & 0 \le t \le T\_{\text{ext}}\\ p(t) &=& \pm \sqrt{(p\_0^2 + \ell\_p^2)e^{-2t} - \ell\_{p'}^2} \end{array} \tag{33}$$

where *TE* is the period given by (we take *E* > 0)

$$T\_E = \cosh^{-1} \frac{E}{2\ell\_x \ell\_p} \to \log \frac{E}{\ell\_x \ell\_p} \quad (E \gg \ell\_x \ell\_p) \,. \tag{34}$$

The asymptotic value of *TE* is the time lapse it takes a particle to go from *x* = *<sup>x</sup>* to *x* = *E*/*<sup>p</sup>* in the *xp* model.

The exchange symmetry *x* ↔ *p* of *xp* is broken by the Hamiltonian (32). To restore it, Berry and Keating proposed the *x* − *p* symmetric Hamiltonian [36]

$$H\_{II} = \left(\mathbf{x} + \frac{\ell\_x^2}{\mathbf{x}}\right) \left(p + \frac{\ell\_p^2}{p}\right), \quad \mathbf{x} \ge \mathbf{0}. \tag{35}$$

Here the classical trajectories turn clockwise around the point (*<sup>x</sup>*, *<sup>p</sup>*), and for *x <sup>x</sup>* and *p p*, approach the parabola *E* = *xp* (see Figure 3). The semiclassical analysis of (32) and (35) reproduce the asymptotic behavior of Equation (2) to leading orders *E* log *E* and *E*, but differ in the remaining terms.

The two models discussed above have the general form

$$H = \mathcal{U}(\mathbf{x})p + \ell\_p^2 \frac{V(\mathbf{x})}{p}, \quad \mathbf{x} \in D,\tag{36}$$

where *U*(*x*) and *V*(*x*) are positive functions defined in an interval *D* of the real line. *HI* corresponds to *U*(*x*) = *V*(*x*) = *x*, *D* = (*<sup>x</sup>*, ∞), and *HI I* corresponds to *U*(*x*) = *V*(*x*) = *x* + -2 *<sup>x</sup>*/*x*, *D* = (0, ∞). The classical Hamiltonian (36) can be quantized in terms of the operator

$$
\hat{H} = \sqrt{\mathcal{U}} \hat{p} \sqrt{\mathcal{U}} + \ell\_p^2 \sqrt{\mathcal{V}} \,\hat{p}^{-1} \sqrt{\mathcal{V}},\tag{37}
$$

where *p*ˆ−<sup>1</sup> is pseudo-differential operator

$$\left(\not p^{-1}\psi\right)(x) = -\frac{i}{\hbar} \int\_{x}^{\infty} dy \,\psi(y),\tag{38}$$

which satisfies that *p*ˆ *p*ˆ−<sup>1</sup> = *p*ˆ−<sup>1</sup> *p*ˆ = **1** acting on functions which vanish sufficiently fast in the limit *<sup>x</sup>* <sup>→</sup> <sup>∞</sup>. The action of *<sup>H</sup>*<sup>ˆ</sup> is

$$(\hat{H}\psi)(\mathbf{x}) = -i\hbar\sqrt{\mathcal{U}(\mathbf{x})}\frac{d}{d\mathbf{x}}\left\{\sqrt{\mathcal{U}(\mathbf{x})}\psi(\mathbf{x})\right\} - \frac{i\ell\_p^2}{\hbar}\int\_{\mathbf{x}}^{\infty} dy \,\sqrt{V(\mathbf{x})V(y)}\,\psi(y). \tag{39}$$

The normal order prescription that leads from (36) to (39) will be derived in Section 7 in the case where *U*(*x*) = *V*(*x*) = *x*, but holds in general [38]. We want the Hamiltonian (37) to be self-adjoint, that is [65,66]

$$
\langle \psi\_1 | \hat{H} | \psi\_2 \rangle = \langle \hat{H} \psi\_1 | \psi\_2 \rangle \,. \tag{40}
$$

When the interval is *D* = (*<sup>x</sup>*, ∞), Equation (40) holds for wave functions that vanishes sufficiently fast at infinity and satisfy the non-local boundary condition

$$
\hbar \, e^{i\theta} \sqrt{\mathcal{U}(\ell\_x)} \, \psi(\ell\_x) = \ell\_p \int\_{\ell\_x}^{\infty} d\mathbf{x} \, \sqrt{V(\mathbf{x})} \, \psi(\mathbf{x}), \tag{41}
$$

where *<sup>ϑ</sup>* <sup>∈</sup> [0, 2*π*) parameterizes the self-adjoint extensions of *<sup>H</sup>*<sup>ˆ</sup> . The quantum Hamiltonian associated with (32) is

$$
\hat{H}\_I = \sqrt{\mathbf{x}} \not\!\!p \sqrt{\mathbf{x}} + \ell\_p^2 \sqrt{\mathbf{x}} \not\!p^{-1} \sqrt{\mathbf{x}}, \qquad \mathbf{x} \ge \ell\_\mathbf{x}. \tag{42}
$$

and its eigenfunctions are proportional to (see Figure 4)

**Figure 4.** Absolute values of the wave function *ψE*(*x*), given in Equation (43) (continuous line), and *x*<sup>−</sup> <sup>1</sup> <sup>2</sup> (dashed line).

$$\psi\_{\rm E}(\mathbf{x}) = \begin{array}{c} \mathbf{x}^{\frac{i\mathbb{E}}{2\hbar}} K\_{\frac{1}{2} - \frac{i\mathbb{E}}{2\hbar}} \left( \frac{\ell\_{p} \mathbf{x}}{\hbar} \right) \propto \begin{cases} \mathbf{x}^{-\frac{1}{2} + \frac{i\mathbb{E}}{\hbar}} & \mathbf{x} \ll \frac{\mathcal{E}}{2\mathcal{E}\_{\mathbf{P}}}\\\mathbf{x}^{-\frac{1}{2} + \frac{i\mathbb{E}}{2\hbar}} \mathbf{e}^{-\ell\_{p} \mathbf{x} / \hbar} & \mathbf{x} \gg \frac{\mathcal{E}}{2\ell\_{\mathbf{P}}} \end{cases} \end{array} \tag{43}$$

where *Kν*(*z*) is the modified *K*-Bessel function [64]. For small values of *x*, the wave functions (43) behave as those of the *xp* Hamiltonian, given in Equation (13), while for large values of *x* they decay exponentially giving a normalizable state. The boundary condition (41) reads in this case

$$
\hbar \, e^{i\theta} \sqrt{\ell\_x} \, \psi(\ell\_x) = \ell\_p \int\_{\ell\_x}^{\infty} d\mathbf{x} \, \sqrt{\mathbf{x}} \, \psi(\mathbf{x}), \tag{44}
$$

and substituting (43) yields the equation for the eigenenergies *En*,

$$e^{i\boldsymbol{\theta}}\mathcal{K}\_{\frac{1}{2}-\frac{i\mathbb{E}}{2\hbar}}\left(\frac{\ell\_x\ell\_p}{\hbar}\right) - K\_{\frac{1}{2}+\frac{i\mathbb{E}}{2\hbar}}\left(\frac{\ell\_x\ell\_p}{\hbar}\right) = 0.\tag{45}$$

For *ϑ* = 0 or *π*, the eigenenergies form time reversed pairs {*En*, −*En*}, and for *ϑ* = 0, there is a zero-energy state *E* = 0. Considering that the Riemann zeros form pairs *sn* = 1/2 ± *itn*, with *tn* real under the RH, and that *s* = 1/2 is not a *zero* of *ζ*(*s*), we are led to the choice *ϑ* = *π*. On the other hand, using the asymptotic behavior

$$\mathcal{K}\_{a+\frac{\vec{w}}{2}}(z) \quad \longrightarrow \quad \sqrt{\frac{\pi}{t}} \left(\frac{t}{z}\right)^a e^{-\pi t/4} e^{\frac{i\pi}{2} \left(a-\frac{1}{2}\right)} \left(\frac{t}{z\varepsilon}\right)^{it/2}, \qquad a > 0, t \gg 1,\tag{46}$$

one derives in the limit |*E*| *h*¯,

$$K\_{\frac{1}{2}+\frac{\tilde{\mu}}{2\hbar}}\left(\frac{\ell\_x \ell\_p}{\hbar}\right) + K\_{\frac{1}{2}-\frac{\tilde{\mu}}{2\hbar}}\left(\frac{\ell\_x \ell\_p}{\hbar}\right) = 0 \longrightarrow \cos\left(\frac{E}{2\hbar}\log\frac{E}{\ell\_x \ell\_p \varepsilon}\right) = 0,\tag{47}$$

hence the number of eigenenergies in the interval (0, *E*) is given asymptotically by

$$m(E) \simeq \frac{E}{2\pi\hbar} \left( \log \frac{E}{\ell\_x \ell\_p} - 1 \right) - \frac{1}{2} + O(E^{-1}).\tag{48}$$

This equation agrees with the leading terms of the semiclassical spectrum (2) and the average Riemann zeros (3) under the identifications (4). Concerning the classical Hamiltonian (35), Berry and Keating obtained, by a semiclassical analysis, the asymptotic behavior of the counting function *n*(*E*)

$$m(t) \simeq \frac{t}{2\pi} \left( \log \frac{t}{2\pi} - 1 \right) - \frac{8\pi}{t} \log \frac{t}{2\pi} + \dots, \quad t \gg 1,\tag{49}$$

where *t* = *E*/¯*h* and *x<sup>p</sup>* = 2*πh*¯. Again, the first two leading terms agree with Riemann's Formula (3), while the next leading corrections are different from (48). In both cases, the constant 7/8 in Riemann's Formula (3) is missing.

**Summary:**

$$\checkmark \text{ The Berry-Keating } xp \text{ model can be implemented quantum mechanics.}$$

✕ The classical *xp* Hamiltonian must be modified with ad-hoc terms to have bounded trajectories.


## **6. The Spacetime Geometry of the Modified** *XP* **Models**

In this section, we show that the modified *xp* Hamiltonian (36) is a disguised general theory of relativity [35]. Let us first consider the Langrangian of the *xp* model,

$$L = p\dot{\mathbf{x}} - H = p\dot{\mathbf{x}} - \mathbf{x}p \,. \tag{50}$$

In classical mechanics, where *H* = *p*2/2*m* + *V*(*x*), the Lagrangian can be expressed solely in terms of the position *x* and velocity *x*˙ = *dx*/*dt*. This is achieved by writing the momentum in terms of the velocity by means of the Hamilton equation *x*˙ = *∂H*/*∂p* = *p*/*m*. However, in the *xp* model the momentum *p* is not a function of the velocity because *x*˙ = *∂H*/*∂p* = *x*. Hence the Lagrangian (50) cannot be expressed uniquely in terms of *x* and *x*˙. The situation changes radically for the Hamiltonian (36) whose Lagrangian is given by

$$L = p\,\dot{\mathbf{x}} - H = p\,\dot{\mathbf{x}} - \mathcal{U}(\mathbf{x})p - \ell\_p^2 \frac{V(\mathbf{x})}{p}.\tag{51}$$

Here the equation of motion

$$
\dot{\mathbf{x}} = \frac{\partial H}{\partial p} = \mathcal{U}(\mathbf{x}) - \ell\_p^2 \frac{V(\mathbf{x})}{p^2},
\tag{52}
$$

allows one to write *p* in terms of *x* and *x*˙,

$$p = \eta \ell\_p \sqrt{\frac{V(\mathbf{x})}{\mathcal{U}(\mathbf{x}) - \dot{\mathbf{x}}}}, \qquad \eta = \text{sign } p \,. \tag{53}$$

where *η* = ±1 is the sign of the momentum that is a conserved quantity. The positivity of *U*(*x*) and *V*(*x*), imply that the velocity *x*˙ must never exceed the value of *U*(*x*). Substituting (53) back into (51), yields the action

$$S\_{\eta} = -\ell\_p \eta \int \sqrt{-ds^2} \,\tag{54}$$

which, for either sign of *η*, is the action of a relativistic particle moving in a 1+1 dimensional spacetime metric

$$ds^2 = 4V(\mathbf{x})(-\mathcal{U}(\mathbf{x})dt^2 + dt d\mathbf{x})\,. \tag{55}$$

The parameter *<sup>p</sup>* plays the role of *mc* where *m* is the mass of the particle and *c* is the speed of light. This result implies that the classical trajectories of the Hamiltonian (36) are the geodesics of the metric (55). The unfamiliar form of (36) is due to a special choice of spacetime coordinates where the component *gxx* of the metric vanishes. A diffeomorphism of *x* permits to set *V*(*x*) = *U*(*x*). The scalar curvature of the metric (55), in this *gauge*, is

$$R(\mathbf{x}) = -2\frac{\partial\_x^2 V(\mathbf{x})}{V(\mathbf{x})},\tag{56}$$

and vanishes for the models *V*(*x*) = *x* and *V*(*x*) = constant. For the Hamiltonian (35) one obtains *R*(*x*) = −4-2 *<sup>x</sup>*/(*x*(*x*<sup>2</sup> + -2 *<sup>x</sup>*)) which vanishes asymptotically.

The flatness of the metric associated with the Hamiltonian (32) implies the existence of coordinates *x*0, *x*<sup>1</sup> where (55) takes the Minkowski form

$$ds^2 = \eta\_{\mu\nu} dx^\mu dx^\nu, \qquad \text{diag } \eta\_{\mu\nu} = \left(-1, 1\right). \tag{57}$$

The change of variables is given by

$$t = \frac{1}{2} \log(\mathbf{x}^0 + \mathbf{x}^1), \qquad \mathbf{x} = \sqrt{- (\mathbf{x}^0)^2 + (\mathbf{x}^1)^2} \,. \tag{58}$$

Let U denote the spacetime domain of the model. In both coordinates it reads

$$\mathcal{M}\_{\varepsilon} = \left\{ (\mathbf{t}, \mathbf{x}) \mid \mathbf{t} \in (-\infty, \infty), \ \mathbf{x} \ge \ell\_{\mathbf{x}} \right\} = \left\{ (\mathbf{x}^{0}, \mathbf{x}^{1}) \mid \mathbf{x}^{0} \in (-\infty, \infty), \ \mathbf{x}^{1} \ge \sqrt{(\mathbf{x}^{0})^{2} + \ell\_{\mathbf{x}}^{2}} \right\}. \tag{59}$$

The boundary of U, denoted by *<sup>∂</sup>*U, is the hyperbola *<sup>x</sup>*<sup>1</sup> = (*x*0)<sup>2</sup> + -2 *<sup>x</sup>*, that passes through the point (*x*0, *x*1)=(0, *<sup>x</sup>*), (see Figure 5).

**Figure 5.** (**Left**): Domain U of Minkowski spacetime given in Equation (59). (**Right**): The classical trajectory given in Equation (33), and plotted in Figure 3-left, becomes a straight line that bounces off regularly at the boundary (dotted line).

A convenient parametrization of the coordinates *x<sup>μ</sup>* is given by the Rindler variables *ρ* and *φ* [67]

$$\mathbf{x}^0 = \rho \sinh \phi, \qquad \mathbf{x}^1 = \rho \cosh \phi,\tag{60}$$

or in light-cone coordinates

$$\mathbf{x}^{\pm} = \mathbf{x}^{0} \pm \mathbf{x}^{1} = \pm \rho \mathbf{c}^{\pm \phi},\tag{61}$$

where the Minkowski metric becomes

$$ds^2 = -d\mathbf{x}^+ \, d\mathbf{x}^- = d\rho^2 - \rho^2 d\phi^2 \,. \tag{62}$$

These coordinates describe the right wedge of Rindler spacetime in 1+1 dimensions

$$\mathcal{R}\_{+} \; : \;= \; \left\{ (\mathbf{x}^{0}, \mathbf{x}^{1}) \mid \mathbf{x}^{0} \in (-\infty, \infty), \; \mathbf{x}^{1} \ge |\mathbf{x}^{0}| \right\} = \left\{ (\boldsymbol{\rho}, \boldsymbol{\phi}) \mid \boldsymbol{\phi} \in (-\infty, \infty), \; \boldsymbol{\rho} > 0 \right\}. \tag{63}$$

Notice that U⊂R+. The boundary *∂*U corresponds to the hyperbola *ρ* = *x* that is the worldline of a particle moving with uniform acceleration equal to 1/*<sup>x</sup>* (in units *c* = 1). The Rindler variables are the ones used to study the Unruh effect [68].

Let us now consider the classical Hamiltonian (35). The underlying metric is given by Equation (55) with *U*(*x*) = *V*(*x*) = *x* + -2 *<sup>x</sup>*/*x*. The change of variables

$$\mathbf{x} = \frac{1}{2}\log(\mathbf{x}^0 + \mathbf{x}^1), \qquad \mathbf{x} = \sqrt{-(\mathbf{x}^0)^2 + (\mathbf{x}^1)^2 - \ell\_{\mathbf{x}}^2} \tag{64}$$

*Symmetry* **2019**, *11*, 494

brings the metric to the form

$$ds^2 = \frac{-(\mathbf{x}^0)^2 + (\mathbf{x}^1)^2}{-(\mathbf{x}^0)^2 + (\mathbf{x}^1)^2 - \ell\_\mathbf{x}^2} \eta\_{\mu\nu} d\mathbf{x}^\mu d\mathbf{x}^\nu = \frac{\rho^2}{\rho^2 - \ell\_\mathbf{x}^2} (d\rho^2 - \rho^2 d\phi^2), \qquad \rho \ge \ell\_\mathbf{x} \,\,\,\tag{65}$$

which in the limit *ρ* → ∞ converges to the flat metric (62).

#### **Summary:**

The classical modified *xp* models are general relativistic theories in 1+1 dimensions. *H* = *x*(*p* + -2 *<sup>p</sup>*/*p*) is related to a domain U of Rindler spacetime. *lp* is the mass of the particle.

1/*<sup>x</sup>* is the acceleration of a particle whose worldline is the boundary of U.

? Relativistic quantum field theory of the modified *xp* models.

#### **7. Diracization of** *H* **=** *X***(***P* **+** *-***2** *<sup>p</sup>***/***P***)**

In this section, we show that the Dirac theory provides the relativistic quantum version of the modified *xp* models [42]. We shall focus on the classical Hamiltonian *H* = *x*(*p* + -2 *<sup>p</sup>*/*p*) because the flatness of the associated spacetime makes the computations easier, but the result is general: the quantum Hamiltonian (39) can be derived from the Dirac equation in a curved spacetime with metric (55) [38].

The Dirac action of a fermion with mass *m* in the spacetime domain (59) is given by (in units *h*¯ = *c* = 1)

$$S = \frac{i}{2} \int\_{\mathcal{U}} d\mathbf{x}^0 d\mathbf{x}^1 \,\bar{\Psi}(\not\partial \!\!/ + im)\psi\,,\tag{66}$$

where *ψ* is a two-component spinor, *ψ*¯ = *ψ*†*γ*0, /*∂* = *γμ∂μ* (*∂μ* = *∂*/*∂xμ*), and *γ<sup>μ</sup>* are the 2d Dirac matrices written in terms of the Pauli matrices *σx*,*<sup>y</sup>* as

$$
\gamma^0 = \sigma^x, \qquad \gamma^1 = -i\sigma^y, \qquad \Psi = \begin{pmatrix} \Psi^- \\ \Psi\_+ \end{pmatrix}. \tag{67}
$$

The variational principle applied to (66) provides the Dirac equation

(/*∂* + *im*)*ψ* = 0 , (68)

and the boundary condition

$$
\dot{\mathbf{x}}^- \psi\_-^\dagger \delta \psi\_- - \dot{\mathbf{x}}^+ \psi\_+^\dagger \delta \psi\_+ = 0,\tag{69}
$$

where *x*˙ ± = *dx*±/*dφ* = *xe*±*<sup>φ</sup>* is the vector tangent to the boundary *<sup>∂</sup>*<sup>U</sup> in the light-cone coordinates *<sup>x</sup>*<sup>±</sup> = *<sup>x</sup>*<sup>0</sup> ± *<sup>x</sup>*1. The Dirac equation reads in components

$$(\partial \!\!\/ - \partial\_1) \psi\_+ + im\psi\_- = 0, \qquad (\partial \!\!\/ + \partial\_1) \psi\_- + im\psi\_+ = 0. \tag{70}$$

If *<sup>m</sup>* <sup>=</sup> 0 then *<sup>ψ</sup>*<sup>±</sup> depends only *<sup>x</sup>*±, and so the fields propagate to the left, *<sup>ψ</sup>*+(*x*+), or to the right, *ψ*−(*x*−), at the speed of light. The derivatives in Equation (70) can be written in terms the variables *t* and *x* using Equation (58),

$$
\partial\_0 - \partial\_1 = -\frac{2e^{2t}}{\chi} \partial\_{\chi}, \qquad \partial\_0 + \partial\_1 = e^{-2t} (\partial\_t + x \partial\_x) \,. \tag{71}
$$

Let us denote by *<sup>ψ</sup>*˜∓(*t*, *<sup>x</sup>*) the fermion fields in the coordinates *<sup>t</sup>*, *<sup>x</sup>* and by *<sup>ψ</sup>*∓(*x*0, *<sup>x</sup>*1) the fields in the coordinates *x*0, *x*1. The relation between these fields is given by the transformation law

$$
\Psi\_- = \left(\frac{\partial \mathbf{x}}{\partial \mathbf{x}^-}\right)^{\frac{1}{2}} \tilde{\Psi}\_- = (2\mathbf{x})^{-\frac{1}{2}} e^t \tilde{\Psi}\_-, \quad \Psi\_+ = \left(\frac{\partial \mathbf{x}}{\partial \mathbf{x}^+}\right)^{\frac{1}{2}} \tilde{\Psi}\_- = (\mathbf{x}/2)^{\frac{1}{2}} e^{-t} \tilde{\Psi}\_+ \,. \tag{72}
$$

Plugging Equations (71) and (72) into (70) gives

$$i\partial\_t \Psi\_- = -i\sqrt{\mathbf{x}} \partial\_x \left(\sqrt{\mathbf{x}} \Psi\_-\right) + m\mathbf{x} \Psi\_+, \qquad \partial\_x \left(\sqrt{\mathbf{x}} \Psi\_+\right) = im\sqrt{\mathbf{x}} \Psi\_- \,. \tag{73}$$

The second equation is readily integrated

$$
\Psi\_{+}(x,t) = -\frac{im}{\sqrt{\infty}} \int\_{x}^{\infty} dy \,\sqrt{y} \Psi\_{-}(y,t) \, , \tag{74}
$$

and replacing it into the first equation in (73) gives

$$i\partial\_t \tilde{\Psi}\_-(\mathbf{x}, t) = -i\sqrt{\mathbf{x}} \partial\_x \left(\sqrt{\mathbf{x}} \tilde{\Psi}\_-\right) - im^2 \sqrt{\mathbf{x}} \int\_x^\infty dy \,\sqrt{y} \,\tilde{\Psi}\_-(y, t) \,. \tag{75}$$

This is the Schrödinger equation with Hamiltonian (42) and the relation *m* = *<sup>p</sup>* found in the previous section. The non-locality of the Hamiltonian (42) is a consequence of the special coordinates *t*, *x* where the component *ψ*˜<sup>+</sup> becomes non-dynamical and depends non-locally on the component *<sup>ψ</sup>*˜<sup>−</sup> that is identified with the wave function of the modified *xp* model. Similarly, the boundary condition (44) can be derived from Equation (69) as follows. In Rindler coordinates the latter equation reads

$$
\varepsilon^{-\phi} \psi\_{-}^{\dagger}(\ell\_{x}, \phi) \; \delta \psi\_{-}(\ell\_{x}, \phi) = \varepsilon^{\phi} \psi\_{+}^{\dagger}(\ell\_{x}, \phi) \; \delta \psi\_{+}(\ell\_{x}, \phi), \quad \forall \phi, \tag{76}
$$

that is solved by

$$-ie^{i\theta}e^{-\phi/2}\,\psi\_{-}(\ell\_{x\prime}\phi) = e^{\phi/2}\,\psi\_{+}(\ell\_{x\prime}\phi), \quad \forall \phi\,,\tag{77}$$

where *ϑ* ∈ [0, 2*π*). Using Equation (72) this equation becomes

$$-ie^{i\theta}\,\tilde{\psi}\_{-}(\ell\_{\mathbf{x}},t)=\tilde{\psi}\_{+}(\ell\_{\mathbf{x}},t),\quad\forall t,\tag{78}$$

that together with Equation (74) yields Equation (44). This completes the derivation of the quantum Hamiltonian and boundary condition associated to *H* = *x*(*p* + -2 *<sup>p</sup>*/*p*). The eigenfunctions and eigenvalue equation of this model were found in Section 5. However, we shall rederive them in alternative way that will provide new insights in the next section.

Let us start by constructing the plane wave solutions of the Dirac Equation (70),

$$
\begin{pmatrix} \psi\_- \\ \psi\_+ \end{pmatrix} \propto \begin{pmatrix} e^{i\pi/4} e^{\beta/2} \\ e^{-i\pi/4} e^{-\beta/2} \end{pmatrix} e^{i(-p^0 \mathbf{x}^0 + p^1 \mathbf{x}^1)} \, \tag{79}
$$

where (*p*0, *p*1) is the energy-momentum vector parameterized in terms of the rapidity variable *β*

$$(p^0)^2 - (p^1)^2 = m^2,\tag{80}$$

$$p^0 = \
i m \sinh \beta, \quad p^1 = \operatorname{im} \cosh \beta, \qquad \beta \in (-\infty, \infty).$$

In Rindler coordinates these plane wave solutions decay exponentially with the distance as corresponds to a localized wave function

$$e^{j(-p^0 \mathbf{x}^0 + p^1 \mathbf{x}^1)} = e^{-m\rho \cosh(\beta - \phi)} \to 0, \quad \text{as} \quad \rho \to \infty. \tag{81}$$

The general solution of the Dirac equation is given by the linear superposition of plane waves (79). The superposition that reproduces the eigenfunctions of the modified *xp* model is

$$\begin{split} \left( \Psi\_{\mp} (\rho, \phi) \right) &= \left. e^{\pm i \pi/4} \int\_{-\infty}^{\infty} d\beta \, e^{-i \mathbb{E}\beta/2} \, e^{\pm \beta/2} \, e^{-m\rho \cosh(\beta - \phi)} \\ &= \left. 2e^{\pm i \pi/4} e^{(\pm \frac{1}{2} - \frac{i\mathbb{E}}{2})\phi} \right. \left. K\_{\frac{1}{2} \mp \frac{i\mathbb{E}}{2}} (m\rho) \right), \end{split} \tag{82}$$

that replaced in Equation (77) gives

$$\epsilon^{i\vartheta} \, \mathcal{K}\_{\frac{1}{2} - \frac{i\mathbb{E}}{2}}(m\ell\_x) - \mathcal{K}\_{\frac{1}{2} + \frac{i\mathbb{E}}{2}}(m\ell\_x) = 0 \,, \tag{83}$$

which coincides with the eigenvalue Equation (45) with *m* = *<sup>p</sup>*. Setting *m<sup>x</sup>* = 2*π* and *ϑ* = *π*, brings Equation (83) to the form

$$\mathcal{G}\_{H}(t) \equiv K\_{\frac{1}{2} + \frac{\tilde{w}}{2}}(2\pi t) + K\_{\frac{1}{2} - \frac{\tilde{w}}{2}}(2\pi) = 0. \tag{84}$$

**Summary:**

The spectrum of a relativistic massive fermion in the domain U agrees with the average Riemann zeros.

? Does this result provide a hint on a physical realization of the Riemann zeros.

#### **8.** *ξ***-Functions: Pólya's Is Massive and Riemann's Is Massless**

The function *ξH*(*t*) appearing in Equation (84) reminds the *fake ξ* function defined by Pólya in 1926 [69,70]

$$\xi^\*(t) = 4\pi^2 \left( K\_{\frac{\theta}{4} + \frac{\tilde{\mu}}{2}}(2\pi) + K\_{\frac{\theta}{4} - \frac{\tilde{\mu}}{2}}(2\pi) \right) . \tag{85}$$

This function shares several properties with the Riemann *ξ* function

$$
\zeta(t) = \frac{1}{4}s(s-1)\Gamma\left(\frac{s}{2}\right)\pi^{-s/2}\zeta(s), \quad s = \frac{1}{2} + it,\tag{86}
$$

namely, *ξ*∗(*t*) is an entire and even function of *t*, its zeros lie on the real axis and behave asymptotically like the average Riemann zeros, as shown by the expansion obtained using Equation (46)

$$\xi^\*(t) \stackrel{t \to \infty}{\longrightarrow} 2^{3/4} \pi^{-7/4} t^{7/4} e^{-\pi t/4} \cos\left(\frac{t}{2} \log\left(\frac{t}{2\pi e}\right) + \frac{7\pi}{8}\right). \tag{87}$$

The zeros of *ξ*(*t*), *ξH*(*t*) and *ξ*∗(*t*) are plotted in Figure 6. The slight displacement between the two top curves is due to the constant 7*π*/8 appearing in the argument of the cosine function in Equation (87) as compared to that in Equation (47).

**Figure 6.** From bottom to top: plot of − log |*ξ*(*t*)| (Riemann zeros), − log |*ξH*(*t*)| (eigenvalues of the Hamiltonian (42) with *x<sup>p</sup>* = 2*π*) and − log |*ξ*∗(*t*)| (Pólya zeros). The cusp represents the zeros of the corresponding functions.

The similarity between *ξH*(*t*) and *ξ*∗(*t*), and the relation between *ξ*∗(*t*) and *ξ*(*t*) provides a hint on the field theory underlying the Riemann zeros. To show this, we shall review how Pólya arrived at *ξ*∗(*t*). The starting point is the expression of *ξ*(*t*) as a Fourier transform [3]

$$\begin{array}{rcl}\xi(t)&=&4\int\_{1}^{\infty}dx\frac{d[\mathbf{x}^{\frac{3}{2}}\psi'(\mathbf{x})]}{d\mathbf{x}}\,\mathbf{x}^{-\frac{1}{4}}\cos\left(\frac{t\log\mathbf{x}}{2}\right),\\\psi(\mathbf{x})&=&\sum\_{n=1}^{\infty}e^{-n^{2}\pi\mathbf{x}},\quad\psi'(\mathbf{x})=\frac{d\psi(\mathbf{x})}{d\mathbf{x}}.\end{array} \tag{88}$$

In the variable *x* = *e<sup>β</sup>* these equations become,

$$\xi(t) = \int\_0^\infty d\beta \, \Phi(\beta) \cos \frac{t\beta}{2},\tag{89}$$

$$\Phi(\beta) = \ 2\pi e^{5\beta/4} \sum\_{n=1}^\infty \left(2\pi e^{\beta} n^2 - 3\right) n^2 e^{-\pi n^2 c^6}.$$

The function Φ(*β*) behaves asymptotically as

$$\Phi(\beta) \to 4\pi^2 e^{9\beta/4} e^{-\pi c^\beta}, \qquad \beta \to \infty,\tag{90}$$

which Pólya replaced by the following expression (see Figure 7).

$$\Phi^\*(\beta) = 4\pi^2 \left( e^{9\beta/4} + e^{-9\beta/4} \right) e^{-\pi(e^{\beta} + e^{-\beta})}.\tag{91}$$

**Figure 7.** Plot of Φ(*β*) (red on line), and Φ∗(*β*) (blue on line). Outside the region |*β*| < 1 the difference is very small.

The function *ξ*∗(*t*) is defined as the Fourier transform of Φ∗(*β*),

$$\xi^\*(t) = \int\_0^\infty d\beta \, \Phi^\*(\beta) \cos \frac{t\beta}{2} \, , \tag{92}$$

which finally gives Equation (85). The function (84) can also be written as the Fourier transform

$$\zeta\_H(t) = \int\_0^\infty d\beta \,\,\Phi\_H(\beta) \cos\frac{t\beta}{2},\tag{93}$$

with

$$\Phi\_H(\beta) = (e^{\oint /2} + e^{-\oint /2})e^{-2\pi \cosh \beta} \tag{94}$$

Observe that the term *e*−2*<sup>π</sup>* cosh *<sup>β</sup>* appears in Φ*H*(*β*) and Φ∗(*β*). The origin of this term in the Dirac theory is the plane wave factor (81) of a fermion with mass *m* located at the boundary *ρ* = *<sup>x</sup>* with *m<sup>x</sup>* = 2*π*. This observation suggests that the Pólya *ξ*<sup>∗</sup> function arises in the relativistic theory of a massive particle with scaling dimension 9/4, rather than 1/2, that corresponds to a fermion (this would explain the different order of the corresponding Bessel functions). The approximation Φ(*β*) Φ∗(*β*), that is *e*−*πe<sup>β</sup> <sup>e</sup>*−2*<sup>π</sup>* cosh *<sup>β</sup>*, can then be understood as the replacement of a massless particle by a massive one. Indeed, the energy-momentum of a massless right moving particle is given by *p*<sup>0</sup> = *p*<sup>1</sup> = Λ*eβ*, where Λ is an energy scale. The corresponding plane wave factor is *e*−*πe<sup>β</sup>* , with Λ = *π*. For large rapidities, *<sup>β</sup>* 1, a massive particle behaves as a massless one, i.e., *<sup>e</sup>*−2*<sup>π</sup>* cosh *<sup>β</sup> <sup>e</sup>*−*πe<sup>β</sup>* . However, for small rapidities this is not the case. These arguments suggest that the field theory underlying the Riemann *ξ* function, if it exists, must associated with a massless particle.

#### **Summary:**

The zeros of the Polya *ξ*∗ function behave as the spectrum of a relativistic massive particle in the domain U.

? Polya's construction of *ξ*∗ suggests that the Riemann's *ξ* function is related to a massless particle.

## **9. The Massive Dirac Model in Rindler Coordinates**

Let us formulate the Dirac theory in Rindler coordinates. Under a Lorentz transformation with boost parameter *λ*, the light-cone coordinates *x*<sup>±</sup> and the Dirac spinors *ψ*<sup>±</sup> transform as

$$\mathbf{x}^{\pm} \to e^{\mp \lambda} \mathbf{x}^{\pm}, \qquad \psi\_{\pm} \to e^{\pm \lambda/2} \,\psi\_{\pm \prime} \tag{95}$$

and the Rindler coordinates (60) as

$$
\phi \to \phi - \lambda, \qquad \rho \to \rho. \tag{96}
$$

Hence the new spinor fields *χ*<sup>±</sup> defined as

$$
\chi\_{\pm} = \mathfrak{e}^{\pm \Phi/2} \Psi\_{\pm \prime} \tag{97}
$$

remain invariant under (96). The Rindler wedge R+, and its domain U, are also invariant under Lorentz transformations. The Dirac action (66) written in terms of the spinors *χ*<sup>±</sup> reads

$$S\_{-}=\,^{i}\int\_{-\infty}^{\infty}d\phi\,\int\_{\ell\_{x}}^{\infty}d\rho\,\left[\chi\_{-}^{\dagger}(\partial\_{\theta}+\rho\partial\_{\rho}+\frac{1}{2})\chi\_{-}+\chi\_{+}^{\dagger}(\partial\_{\theta}-\rho\partial\_{\rho}-\frac{1}{2})\chi\_{+}+im\rho\,\left(\chi\_{-}^{\dagger}\chi\_{+}+\chi\_{+}^{\dagger}\chi\_{-}\right)\right],\tag{98}$$

while the Dirac Equation (68) and the boundary condition (77) become

$$(\partial\_{\Phi} \pm \rho \partial\_{\rho} \pm \frac{1}{2}) \chi\_{\mp} + im\rho \chi\_{\pm} = 0,\tag{99}$$

and

$$-ie^{i\theta}\,\chi\_{-}=\chi\_{+}\quad\text{at}\ \rho=\ell\_{X}\,.\tag{100}$$

The infinitesimal generator of translations of the Rindler time *φ*, acting on the spinor wave functions, is the Rindler Hamiltonian *HR*, which can be read off from (99)

$$i\partial\_{\phi}\chi = H\_{\mathbb{R}}\chi\_{\prime} \qquad \chi = \begin{pmatrix} \chi\_{-} \\ \chi\_{+} \end{pmatrix},\tag{101}$$

$$H\_R = \begin{pmatrix} -i(\rho \,\partial\_{\rho} + \frac{1}{2}) & m\rho\\ m\rho & i(\rho \,\partial\_{\rho} + \frac{1}{2}) \end{pmatrix} = \sqrt{\rho} \,\not p\_{\rho} \sqrt{\rho} \,\sigma^z + m\rho \,\sigma^x,\tag{102}$$

where *p*ˆ*<sup>ρ</sup>* = −*i∂*/*∂ρ*, is the momentum operator conjugate to the radial coordinate *ρ*. Notice that the operator

$$H\_{\rho\overline{\rho}\_{\overline{\rho}}} = -i(\rho\,\partial\_{\overline{\rho}} + \frac{1}{2}) = \frac{1}{2}(\rho\,\hat{p}\_{\overline{\rho}} + \hat{p}\_{\overline{\rho}}\rho) = \sqrt{\rho}\,\hat{p}\_{\overline{\rho}}\sqrt{\rho},\tag{103}$$

coincides with Equation (11) with the identification *x* = *ρ* (in units *h*¯ = 1). The eigenfunctions of (103) are

$$H\_{\rho p\_{\theta}} \, \psi\_{E} = E \, \psi\_{E} \qquad \psi\_{E} = \frac{1}{\sqrt{2\pi}} \rho^{-1/2 + iE} \,, \tag{104}$$

with real eigenvalue *E* for *ρ* > 0 (recall Equation (13)). Thus, *HR* consists of two copies of *xp*, with different signs corresponding to opposite fermion chiralities that are coupled by the mass term *mρσx*.

The scalar product of two wave functions, in the domain U, can be defined as

$$
\langle \chi\_1 | \chi\_2 \rangle = \int\_{\ell\_x}^{\infty} d\rho \left( \chi\_{1,-}^\* \chi\_{2,-} + \chi\_{1,+}^\* \chi\_{2,+} \right) \,. \tag{105}
$$

The Hamiltonian *HR* is Hermitian with this scalar product acting on wave functions that satisfy Equation (100) and vanish sufficiently fast at infinity, i.e., lim*ρ*→<sup>∞</sup> *<sup>ρ</sup>*1/2*χ*±(*ρ*, *<sup>φ</sup>*) = 0. The eigenvalues and eigenvectors of the Hamiltonian (102), are given by the solutions of the Schrödinger equation

$$H\_{\mathbb{R}}\chi = E\_{\mathbb{R}}\chi,\qquad \chi\_{\pm}(\rho,\phi) = e^{-iE\_{\mathbb{R}}\phi \mp i\pi/4}K\_{\frac{1}{\mathbb{Z}}\pm iE\_{\mathbb{R}}}(m\rho), \qquad \rho \ge \ell\_{\ge} \tag{106}$$

which coincide with Equation (82) with the identification

$$E\_R = \frac{E}{2} \,. \tag{107}$$

The factor of 1/2 comes from the relation *e*2*<sup>t</sup>* = *x*<sup>0</sup> + *x*<sup>1</sup> = *ρe<sup>φ</sup>* (see Equation (58)), that implies *e*−*iER<sup>φ</sup>* ∝ *e*−*iEt*. The Rindler eigenenergies are obtained replacing *E* by 2*ER* in Equation (83).

## **Comments:**

• The Dirac Hamiltonian associated with the metric (65) is

$$H = \begin{pmatrix} h & m\rho\Lambda \\ m\rho\Lambda & -h \end{pmatrix}, \qquad h = -i\left(\rho\partial\_{\rho} + \frac{1}{2} + \frac{1}{2}\rho\partial\_{\theta}(\log\Lambda)\right), \qquad \Lambda = \frac{\rho}{\sqrt{\rho^2 - \ell\_X^2}}.\tag{108}$$

In the limit *ρ x* this Hamiltonian converges towards (102).

• Gupta, Harikumar and de Queiroz proposed the Hamiltonian (*x*/*p* + /*px*)/2 as a Dirac variant of the *xp* Hamiltonian [37]. The Hamiltonian is defined on a semi-infinite cylinder and effectively becomes one dimensional by considering the winding modes on the compact dimension. The eigenfunctions are given by Whittaker functions and the spectrum satisfies an equation similar to Equation (29) in the Landau theory. In the limit where a regularization parameter goes to zero one obtains a continuum spectrum with a correction term related to the Riemann–von Mangoldt formula.

• Bender, Brody and Müller proposed recently a generalization of the *xp* operator [43]

$$H = \frac{\mathbf{1}}{\mathbf{1} - e^{-i\hat{\boldsymbol{\beta}}}} (\mathbf{x}\boldsymbol{\hat{\rho}} + \boldsymbol{\hat{\rho}}\mathbf{x}) (\mathbf{1} - e^{-i\hat{\boldsymbol{\rho}}}) ,\tag{109}$$

with the property that its eigenvalues *En* give the Riemann zeros as *zn* = <sup>1</sup> <sup>2</sup> (1 − *iEn*). This interesting result follows from the fact the eigenfunctions of (109) are given in terms of the Hurwitz zeta function as *ψz*(*x*) = *ζ*(*z*, *x* + 1) and imposing the boundary condition

$$
\psi\_{\mathbb{Z}\_n}(0) = 0 \to \mathbb{Z}(z\_{\mathbb{R}}, 1) = \mathbb{Z}(z\_{\mathbb{R}}) = 0. \tag{110}
$$

Unfortunately, the operator (109) is not self-adjoint, so that the reality of its eigenvalues is not guaranteed. However, the authors of [43] found that *iH* has a *PT* symmetry which, if it is maximally broken, would imply the reality of the eigenvalues. This property though remains to be proved. Further details can be found in references [44,45].

#### **Summary:**

The massless Dirac Hamiltonian in Rindler spacetime is the direct sum of *xp* and −*xp*. The mass term couples the left and right modes of the fermions.

#### **10. The Massless Dirac Equation with Delta Function Potentials**

From analogies between the Polya *ξ*∗ function, the Riemann *ξ* function and the *ξ<sup>H</sup>* function of the massive Dirac model, we conjectured in Section 8 the existence of a massless field theory underlying *ξ*. At first look this idea does not look correct because the Hamiltonian obtained by setting *m* = 0 in Equation (102), is equivalent to two copies of the quantum *xp* model which has a continuum spectrum. In fact, the mass term in that Hamiltonian is the mechanism responsible for obtaining a discrete spectrum.

To resolve this puzzle, we shall replace the *bulk* mass term in the Dirac action (98) by a sum of ultra-local interactions placed at fixed values *<sup>n</sup>* of the radial coordinate *ρ* [42]. These interactions can arise from moving mirrors, or beam splitters, that move with a uniform acceleration 1/*n* (see Figure 8). The fermion moves freely, until it hits one of the mirrors and it is reflected or transmitted. The moving mirrors are realized mathematically by delta functions added to the massless Dirac action that couple the left and right components of the fermion on both sides of the mirror. These delta functions provide the matching conditions for the wave functions and can be parameterized by a complex number *<sup>n</sup>* with *n* = 2, ... , ∞. The scattering of the fermion at each mirror preserves unitarity that is equivalent to the self-adjointness of the Hamiltonian.

**Figure 8.** (**Left**): worldlines of the mirrors with accelerations *an* = 1/*<sup>n</sup>* = 1/*n* (*n* = 1, 2, ...). (**Right**): A massless fermion (dotted line) at the point (*x*0, *x*1)=(0, 1) moves to the right until it hits a moving mirror where it can be reflected or transmitted.

The model is formulated in the spacetime U defined in Equation (59). We divide U into an infinite number of domains separated by hyperbolas with constant values of *ρ* = *n*, as follows. First we define the intervals (see Figure 9)

$$I\_{\mathbb{N}} = \{ \rho \mid \ell\_{\mathbb{N}} < \rho < \ell\_{n+1} \}, \quad n = 1, 2, \ldots, \infty,\tag{111}$$

where using the scale invariance of the model we set -<sup>1</sup> = 1 (-<sup>1</sup> plays the role of *x* in previous sections).

**Figure 9.** Intervals *In* defined in Equation (111).

The partition of U is given by

$$\mathcal{U} \quad \rightarrow \quad \tilde{\mathcal{U}} = \cup\_{n=1}^{\infty} \mathcal{U}\_{\mathfrak{n}} \qquad \mathcal{U}\_{\mathfrak{n}} = \mathcal{Z}\_{\mathfrak{n}} \times \mathbb{R} \,. \tag{112}$$

where the factor R denotes the range of the Rindler time *φ*. See Figure 8 for an example with *<sup>n</sup>* = *n*. The wave function of the model is the two component Dirac spinor (see Equation (101))

$$\chi(\rho) = \begin{pmatrix} \chi\_{-}(\rho) \\ \chi\_{+}(\rho) \end{pmatrix}, \qquad \rho \in \mathcal{Z} = \cup\_{n=1}^{\infty} I\_{n\prime} \tag{113}$$

and the scalar product is given by (recall Equation (105))

$$
\langle \chi | \chi \rangle \quad = \sum\_{n=1}^{\infty} \int\_{\ell\_n}^{\ell\_{n+1}} d\rho \, \chi^\dagger(\rho) \cdot \chi(\rho). \tag{114}
$$

The complex Hilbert space is H = *<sup>L</sup>*2(I, C) ⊕ *<sup>L</sup>*2(I, C) and the Hamiltonian is obtained setting *m* = 0 in Equation (102)

$$H = \begin{pmatrix} -i(\rho \,\partial\_{\rho} + \frac{1}{2}) & 0\\ 0 & i(\rho \,\partial\_{\rho} + \frac{1}{2}) \end{pmatrix}, \quad \rho \notin \mathcal{T} \,. \tag{115}$$

*H* is a self-adjoint operator acting on the subspace H*<sup>ϑ</sup>* ⊂ H of wave functions that satisfy the boundary conditions [42] (see [71] for the relation between self-adjointness of operators and boundary conditions)

$$\chi\_{\pi} \in \, \, \mathcal{H}\_{\theta} \colon \quad \chi(\ell\_{\mathfrak{n}}^{-}) = L(\varrho\_{\mathfrak{n}}) \chi(\ell\_{\mathfrak{n}}^{+}), \quad (n \ge 2), \quad -ie^{i\theta} \, \chi\_{-}(\ell\_{1}^{+}) = \chi\_{+}(\ell\_{1}^{+}), \tag{116}$$

where

$$\chi(\ell\_n^{\pm}) = \lim\_{\varepsilon \to 0^{+}} \chi(\ell\_n \pm \varepsilon),\tag{117}$$

and

$$\vartheta \in [0, 2\pi), \qquad L(\varrho) \quad = \begin{array}{c} 1 \\ \hline 1 - |\varrho|^2 \end{array} \begin{pmatrix} 1 + |\varrho|^2 & 2i\varrho \\ -2i\varrho^\* & 1 + |\varrho|^2 \end{pmatrix}, \quad \varrho \in \mathbb{C}, \quad |\varrho| \neq 1. \tag{118}$$

This means that *H* satisfies

$$
\langle \chi\_1 | H \chi\_2 \rangle = \langle H \chi\_1 | \chi\_2 \rangle, \qquad \chi\_{1,2} \in \mathcal{H}\_\vartheta. \tag{119}
$$

This condition guarantees that the norm (114) of the state is conserved by the time evolution generated by the Hamiltonian. The subspace H*<sup>ϑ</sup>* also depends on *<sup>n</sup>* and *<sup>n</sup>* but we shall not write this dependence explicitly. Similarly, we shall also denote the Hamiltonian as *Hϑ*. The matching conditions (116) describe a scattering process where two incoming waves *χ*in *<sup>n</sup>* collide at the *n*th-mirror and become two outgoing waves *χ*out *<sup>n</sup>* given by (see Figure 10)

$$\chi\_n^{\rm in} = \begin{pmatrix} \chi\_- (\ell\_n^-) \\ \chi\_+ (\ell\_n^+) \end{pmatrix}, \qquad \chi\_n^{\rm out} = \begin{pmatrix} \chi\_- (\ell\_n^+) \\ \chi\_+ (\ell\_n^-) \end{pmatrix}, \quad n > 1. \tag{120}$$

At the mirror *n* = 1, the components *χ*±(-− <sup>1</sup> ) of these vectors are null, i.e., there is no propagation at the left of the boundary. The scattering process is described by the matrix *Sn*

$$\chi\_n^{\rm out} = S\_n \chi\_n^{\rm in}, \qquad S\_n = \frac{1}{1 + |\varrho\_n|^2} \begin{pmatrix} 1 - |\varrho\_n|^2 & -2i\varrho\_n \\ -2i\varrho\_n^\* & 1 - |\varrho\_n|^2 \end{pmatrix}, \quad n > 1,\tag{121}$$

that is unitary,

$$S\_{\text{ll}} S\_{\text{n}}^{\dagger} = \mathbf{1} \,. \tag{122}$$

Notice that the boundary condition at *ρ* = -1, is also described by Equation (121) with a parameter <sup>1</sup>

$$
\varrho\_1 = -e^{-i\theta} \, , \tag{123}
$$

that is a pure phase for the Hamiltonian *H<sup>ϑ</sup>* to be self-adjoint. The matrix *L*() satisfies

$$L(1/\varrho^\*) = -L(\varrho). \tag{124}$$

Hence, replacing *<sup>n</sup>* by 1/∗ *<sup>n</sup>* gives a unitary equivalent model because the sign changes at *ρ* = *n*, given in Equation (124), can be compensated by changing the sign of the wave function in the remaining intervals. Hence, without losing generality, we shall impose the condition |*n*| < 1, ∀*n* > 1.

**Figure 10.** (**Top**): scattering process taking place at the mirror located at *ρ* = *<sup>n</sup>* for *n* > 1 (Equation (121)). (**Bottom**): reflexion at the perfect mirror located at *ρ* = -<sup>1</sup> (Equation (116)).

The eigenfunctions of the Hamiltonian (115) are the customary functions (see Equation (13))

$$H\,\chi = E\,\chi \longrightarrow \chi\_{\mp} \propto \rho^{-1/2\pm iE}.\tag{125}$$

From now one, we shall assume that *E* is a real number which is guaranteed by the self-adjointness of the Hamiltonian *H*. In the *n*th interval we take

$$\chi\_{\mp,n}(\rho) = e^{\pm i\pi/4} \frac{A\_{\mp,n}}{\rho^{1/2 \mp iE}}, \qquad \ell\_n < \rho < \ell\_{n+1}.\tag{126}$$

where *<sup>A</sup>*∓,*<sup>n</sup>* are constants that in general will depend on *<sup>E</sup>*. The phases *<sup>e</sup>*±*iπ*/4 have been introduced by analogy with those appearing in Equation (106). The boundary values of *χ* at *ρ* = -± *<sup>n</sup>* (*n* ≥ 1) are (see Equation (117))

$$\chi\_{\mp}(\ell\_n^+) = \chi\_{\mp,n}(\ell\_n) = \varepsilon^{\pm \frac{i\pi}{4}} \frac{A\_{\mp,n}}{\ell\_n^{1/2 \mp iE}}, \qquad \chi\_{\mp}(\ell\_n^-) = \chi\_{\mp,n-1}(\ell\_n) = \varepsilon^{\pm \frac{i\pi}{4}} \frac{A\_{\mp,n-1}}{\ell\_n^{1/2 \mp iE}}.\tag{127}$$

Let us define the vectors

$$|\mathbf{A}\_{\mathbb{N}}\rangle = \begin{pmatrix} A\_{-,\mathbb{N}} \\ A\_{+,\mathbb{N}} \end{pmatrix}, \qquad n \ge 1. \tag{128}$$

The boundary conditions (116) together with Equation (127) imply

$$|\mathbf{A}\_{n-1}\rangle \quad = \, \, T\_n |\mathbf{A}\_n\rangle \quad (n \ge 2), \qquad |\mathbf{A}\_1\rangle = |\mathbf{A}\_1(\theta)\rangle = \left(\begin{array}{c} 1 \\ e^{i\theta} \end{array}\right) ,\tag{129}$$

where the transfer matrix *Tn* is given by

$$T\_{\rm nl} = \frac{1}{1 - |\varrho\_n|^2} \begin{pmatrix} 1 + |\varrho\_n|^2 & 2\varrho\_n \ell\_n^{-2i\mathcal{E}}\\ 2\varrho\_n^\* \ell\_n^{2i\mathcal{E}} & 1 + |\varrho\_n|^2 \end{pmatrix} \quad (n \ge 2). \tag{130}$$

The norm of the eigenstate can be computed using Equations (114) and (126)

$$||\chi||^2 = \sum\_{n=1}^{\infty} \log \frac{\ell\_{n+1}}{\ell\_n} \left< \mathbf{A}\_n | \mathbf{A}\_n \right> , \qquad \left< \mathbf{A}\_n | \mathbf{A}\_n \right> = |A\_{-,n}|^2 + |A\_{+,n}|^2. \tag{131}$$

The log term comes from the integral of the norm of the wave function in the *n*th interval, *n*+1 *<sup>n</sup> <sup>d</sup>ρ*/*<sup>ρ</sup>* (we used that *<sup>E</sup>* is real). If *<sup>n</sup>* = 0 then *Tn* = **<sup>1</sup>** which implies that |**A***n*−1 = |**A***n*. If this happens for all *n*, then |**A***n* = |**A**1, in which case the norm of these states diverges, but they can be normalized using Dirac delta functions, so they correspond to scattering states. In the general case, iterating Equation (129) yields |**A***n* in terms of |**A**1(*ϑ*)

$$|\mathbf{A}\_{\rm n}\rangle\_{\cdot} = \,^{T}T\_{n}^{-1}T\_{n-1}^{-1}\cdots T\_{2}^{-1}|\mathbf{A}\_{1}(\theta)\rangle, \quad n \ge 2\,. \tag{132}$$

For special values of *<sup>n</sup>* and *<sup>n</sup>* one can find the exact expression of these amplitudes. An example is *<sup>n</sup>* = *e<sup>n</sup>*/2, *<sup>n</sup>* = cte [42]. To make contact with the Riemann zeros, we shall consider a limit where the reflection coefficients vanish asymptotically.

#### **Summary:**

The massless Dirac Hamiltonian with delta function potential is solvable by transfer matrix methods.

The model is completely characterized by the set of parameters {*<sup>n</sup>*, *n*}<sup>∞</sup> *<sup>n</sup>*=<sup>2</sup> and *ϑ*.

#### **11. Heuristic Approach to the Spectrum**

Let us replace *<sup>n</sup>* by *εn*, and consider the limit *ε* → 0 of the transfer matrix (130)

$$T\_n \quad \simeq \quad \mathbf{1} + \varepsilon \tau\_n + O(\varepsilon^2), \qquad \tau\_n = \begin{pmatrix} 0 & 2\varrho\_n \ell\_n^{-2iE} \\ 2\varrho\_n^\* \ell\_n^{2iE} & 0 \end{pmatrix} \qquad (n \ge 2). \tag{133}$$

Plugging this equation into Equation (132) yields

$$|\mathbf{A}\_{\mathfrak{N}}\rangle \simeq \left(\mathbf{1} - \varepsilon \sum\_{m=2}^{n} \tau\_{\mathfrak{m}}\right) |\mathbf{A}\_{1}(\mathfrak{\theta})\rangle + O(\varepsilon^{2}), \quad n \ge 2,\tag{134}$$

and in components

$$A\_{-,\mu} \quad \simeq \quad 1 - 2\varepsilon e^{i\theta} \sum\_{m=2}^{n} \varrho\_{m} \ell\_{m}^{-2i\mathbb{E}} + O(\varepsilon^{2}), \qquad A\_{+,\mu} \simeq e^{i\theta} - 2\varepsilon \sum\_{m=2}^{n} \varrho\_{m}^{\*} \ell\_{m}^{2i\mathbb{E}} + O(\varepsilon^{2}).\tag{135}$$

For a normalizable state, the amplitudes *A*±,*<sup>n</sup>* must vanish as *n* → ∞. In the next section we shall study in detail the normalizability of the state. We shall make the following choice of lengths and reflection coefficients [42]

$$\ell\_n = n^{1/2}, \qquad \varrho\_n = \frac{\mu(n)}{n^{1/2}}, \qquad n > 1,\tag{136}$$

where *<sup>μ</sup>*(*n*) is the Moëbius function that is equal to (−1)*<sup>r</sup>* , with *r* the number of distinct primes factors of a square free integer *n*, and *μ*(*n*) = 0, if *n* is divisible by the square of a prime number [4]. See Figures 11 and 12 for a graphical representation of Equations (136) and (135). The Moebius function has been used in the past to provide physical models of prime numbers, most notably in the ideal gas of primons with fermionic statistics [72,73] and a potential whose semiclassical spectrum are the primes [46,74].

**Figure 11.** Localization of the mirrors corresponding to the choice (136), together with the values of *μ*(*n*).

**Figure 12.** Depiction of the amplitudes *A*±,<sup>∞</sup> as the superposition of a principal wave with the waves resulting from the scattering with all the mirrors along its trajectory (see Equation (135)). The terms of higher order in *ε* correspond to more than one scattering.

Another motivation of the choice (136) is the following [42]. Consider a fermion that leaves the boundary at *ρ* = -1, moves rightwards until it hits the mirror at *ρ* = *<sup>n</sup>* where it gets reflected and returns to the boundary. The time lapse for the entire trajectory is given by

$$
\pi\_n = 2\log(\ell\_n/\ell\_1) \tag{137}
$$

where we used the Rindler metric Equation (62). If the mirror is associated with the prime *p*, that is *<sup>p</sup>* = √*p*, the time will be given by *<sup>τ</sup><sup>p</sup>* = log *<sup>p</sup>*. This result reminds the Berry conjecture that postulates the existence of a classical chaotic Hamiltonian whose primitive periodic orbits are labelled by the primes *p*, with periods log *p*, and whose quantization will give the Riemann zeros as energy levels [12]. A classical Hamiltonian with this property has not been found, but the array of mirrors presented above, displays some of its properties. In particular, the trajectory between the boundary and the mirror at *<sup>p</sup>*, with *p* a prime number, behaves as a primitive orbit with a period log *p*. Moreover, the trajectories and periods of these orbits are independent of the energy of the fermion because it moves at the speed of light.

Let us work out the consequences (136). The condition for a normalizable eigenstate, that is lim*n*→<sup>∞</sup> *A*±,*<sup>n</sup>* = 0, is

$$1 \simeq 2\varepsilon \, e^{i\theta} \sum\_{n=1}^{\infty} \frac{\mu(n)}{n^{\frac{1}{2} + iE}} = \frac{2\varepsilon \, e^{i\theta}}{\zeta(\frac{1}{2} + iE(\varepsilon))},\tag{138}$$

where we have included the term *n* = 1 in the series because it does not modify its value when *ε* → 0. We have employed the formula ∑<sup>∞</sup> *<sup>n</sup>*=<sup>1</sup> *μ*(*n*)/*n<sup>s</sup>* = 1/*ζ*(*s*) for a value of *s* where the series may not converge. In the next section we shall compute the value of the finite sum that determines the norm of the state. *En*(*ε*) denotes a solution such that lim*ε*→<sup>0</sup> *En*(*ε*) = *En*, where <sup>1</sup> <sup>2</sup> + *iEn* is a zero of the zeta function. All known zeros of *ζ*(*s*) on the critical line are simple, but we shall also consider the case

where <sup>1</sup> <sup>2</sup> <sup>+</sup> *iEn* might be a zero of order *<sup>r</sup>* <sup>≥</sup> 1, that is *<sup>ζ</sup>*(*r*)(*s*) <sup>=</sup> 0. The Taylor expansion of *<sup>ζ</sup>*( <sup>1</sup> <sup>2</sup> + *iE*(*ε*)) around <sup>1</sup> <sup>2</sup> + *iEn*, in Equation (138) yields

$$1 \simeq \frac{2\varepsilon \, r! \, e^{i\vartheta}}{i^r (E\_n(\varepsilon) - E\_n)^r \zeta^{(r)}(\frac{1}{2} + iE\_n)}.\tag{139}$$

Hence *En*(*ε*) − *En* is of order *<sup>ε</sup>*1/*<sup>r</sup>* , as *ε* → 0 and

$$\frac{\zeta^{(r)}(\frac{1}{2} + iE\_n)}{\zeta^{(r)}(\frac{1}{2} - iE\_n)} = (-1)^r e^{2i\vartheta} \,. \tag{140}$$

On the other hand, from Equation (9) one finds

$$i^r \zeta^{(r)}(\frac{1}{2} + iE\_n) = e^{-i\theta(E\_n)} Z^{(r)}(E\_n) \,. \tag{141}$$

that plugged into (140) yields

$$e^{2i(\theta + \theta(E\_n))} = 1, \qquad \forall r \,. \tag{142}$$

We can collect these results in the equation

$$\text{If } \sqrt{\zeta}(\frac{1}{2} \pm iE\_n) = 0 \quad \text{and} \quad e^{2i(\vartheta + \theta(E\_n))} = 1 \Longleftrightarrow H\_\theta \chi\_{E\_n} = E\_n \chi\_{E\_n}.\tag{143}$$

Observe that *ϑ* is fixed mod *π*. In the next section we shall fix this ambiguity. This equation is heuristic. It has been derived by (i) solving the eigenvalue equation in the limit *ε* → 0, (ii) imposing the vanishing of the eigenfunction at infinity and (iii) using the Dirichlet series of 1/*ζ*(*s*) in a region where it may not converge. In the next section we shall derive Equation (143) without making the previous assumptions (see Equation (176)). Let us notice that this spectral realization of the *zeros* requires the fine tuning of the parameter *ϑ* in terms of the phase of the zeta function, *θ*(*En*) (see Figure 13). This realization is different from the Pólya-Hilbert conjecture of a single Hamiltonian encompassing all the Riemann zeros at once. This Hamiltonian would exist if *θ*(*En*) = *θ*0, ∀*n*, but this is certainly not the case.

**Figure 13.** Schematic representation of the array of mirrors that give rise to a spectral realization of the Riemann zeros. The red and blue lines represent the left and right wave functions *χ*±,*n*(*ρ*). The wave functions are discontinuous at the moving mirrors located at the positions *<sup>n</sup>* <sup>=</sup> <sup>√</sup>*<sup>n</sup>* with *<sup>n</sup>* a square free integer. The knob on the left represents the scattering phase at the perfect mirror that is set to minus the phase of the zeta function at the *zero En*, namely *ϑ* = −*θ*(*En*) mod *π*.

#### **Summary:**

A Riemann zero, on the critical line, becomes an eigenvalue of the Hamiltonian *H<sup>ϑ</sup>* by tuning the phase *ϑ* according to the phase of the zeta function.

✕ The previous result is obtained in the limit *ε* → 0 and is heuristic.

#### **12. The Riemann Zeros as Spectrum and the Riemann Hypothesis**

In this section, we provide more rigorous arguments that support the heuristic results obtained previously. Let us first review the main properties of the model discussed so far. The Hamiltonian, Equation (115), describes the dynamics of a massless Dirac fermion in the region of Rindler spacetime bounded by the hyperbola *ρ* = -1. The reflection of the wave function at this boundary is characterized by a parameter *ϑ*, which is real for a self-adjoint Hamiltonian. At the positions *<sup>n</sup>*><sup>1</sup> the wave function is discontinuous due to the presence of delta function potentials characterized by the reflection amplitudes *<sup>n</sup>* that provide the matching conditions of the wave function at those sites. An eigenfunction *χ*, with eigenvalue *E*, has a simple expression, Equation (126), in terms of the amplitudes *An*,±, which are related by the transfer matrix *Tn* (130). The norm of *χ* is given by the sum of the squared length of the vectors **A***n*, weighted with a factor that depends on the positions *n*, Equation (131). We introduce a scale factor *ε* in the parameters *n*, which allows us to study the limit *ε* → 0, where the mirrors become semitransparent. In this way we found an ansatz for the parameters *<sup>n</sup>* and *<sup>n</sup>* that heuristically led to an individual spectral realization of the *zeros* by fine tuning the parameter *ϑ*.

#### *12.1. Normalizable Eigenstates*

Under the choice *<sup>n</sup>* = *n*1/2, Equation (131) becomes

$$||\chi||^2 = \frac{1}{2} \sum\_{n=1}^{\infty} \log\left(1 + \frac{1}{n}\right) \left< \mathbf{A}\_{\mathbb{H}} | \mathbf{A}\_{\mathbb{H}} \right>\,. \tag{144}$$

This series can be replaced by

$$||\chi||\_c^2 \equiv \sum\_{n=1}^{\infty} \frac{1}{n} \left< \mathbf{A}\_n | \mathbf{A}\_n \right>\,\tag{145}$$

which is convergent if and only if (144) is convergent. The vectors **A***<sup>n</sup>* are obtained by acting on **A**1(*ϑ*) with the transfer matrices *Tn* (see Equation (132)). These matrices have unit determinant and can be written as the exponential of traceless Hermitian matrices, that is,

$$T\_{\mathbb{H}} = e^{\tau\_{\mathbb{H}}}, \qquad \tau\_{\mathbb{H}} = \begin{pmatrix} 0 & r\_n \ell\_n^{-2iE} \\ r\_n^\* \ell\_n^{2iE} & 0 \end{pmatrix}, \qquad \forall E \in \mathbb{R}\_{\prime} \tag{146}$$

where taking |*n*| < 1,

$$r\_n = \frac{\varrho\_n}{|\varrho\_n|} \log \frac{1 + |\varrho\_n|}{1 - |\varrho\_n|}, \qquad \varrho\_n = \frac{r\_n}{|r\_n|} \tanh \frac{|r\_n|}{2} \,. \tag{147}$$

To derive Equation (146) we used the relation

$$\exp\left(\begin{array}{cc} 0 & a\\ b & 0 \end{array}\right) = \begin{pmatrix} \cosh(\sqrt{ab}) & \frac{a}{\sqrt{ab}}\sinh(\sqrt{ab})\\ \frac{b}{\sqrt{ab}}\sinh(\sqrt{ab}) & \cosh(\sqrt{ab}) \end{pmatrix}, \quad \forall a, b \in \mathbb{C} - \{0\}. \tag{148}$$

If |*n*| << 1 one gets *rn* 2*n*, hence in that limit both parameters give the same result. Using Equation (146), the recursion relation (132) reads

$$|\mathbf{A}\_k\rangle = e^{-\tau\_k} e^{-\tau\_{k-1}} \dots e^{-\tau\_2} |\mathbf{A}\_1\rangle, \quad k \ge 2. \tag{149}$$

#### *12.2. The Magnus Expansion*

It is rather difficult to find an analytic expression of the product of matrices of Equation (149). However, we can estimate it replacing *rn* by *εrn*, and taking the limit *ε* → 0. Under this replacement Equation (149) becomes

$$|\mathbf{A}\_k\rangle = e^{-\varepsilon\tau\_k} e^{-\varepsilon\tau\_{k-1}} \dots e^{-\varepsilon\tau\_2} |\mathbf{A}\_1\rangle \quad (k \ge 2). \tag{150}$$

The product of exponentials of matrices can be expressed as the exponential of a matrix given by the Magnus expansion [75]

$$e^{-\varepsilon\tau\_k}e^{-\varepsilon\tau\_{k-1}}\dots e^{-\varepsilon\tau\_2} = \exp\left(-\varepsilon\sum\_{n=2}^{n}\tau\_n - \frac{\varepsilon^2}{2}\sum\_{n\_1>n\_2=2}^{k}[\tau\_{n\_1},\tau\_{n\_2}] + O(\varepsilon^3)\right) \quad (k \ge 2). \tag{151}$$

In the limit *ε* → 0 we truncate this expression to the term of order *ε*,

$$\varepsilon^{-\varepsilon\tau\_{k}}\varepsilon^{-\varepsilon\tau\_{k-1}}\dots\varepsilon^{-\varepsilon\tau\_{2}} \simeq \exp\left(\begin{array}{c} 0 \\ -\varepsilon\sum\_{n=2}^{k}r\_{n}^{\*}\ell\_{n}^{\Delta\overline{E}} \end{array}\right) \simeq \exp\left(\begin{array}{c} 0 \\ -\varepsilon\Lambda\_{z}^{\*}(k) \end{array}\right),\tag{152}$$

which using (136)

$$r\_n = \frac{\mu(n)}{n^{1/2}}\tag{153}$$

gives

$$M\_z(k) = 1 + \sum\_{n=2}^{k} r\_n \ell\_n^{-2iE} = \sum\_{n=1}^{k} \frac{\mu(n)}{n^z}, \qquad z = \frac{1}{2} + iE \,. \tag{154}$$

We have added the constant 1 to *Mz*(*k*), which does not affect the results in the limit *ε* → 0 . Using Equations (148), (150) and (152) we obtain

<sup>|</sup>**A***n* exp 0 −*εMz* (*n*) −*εM*<sup>∗</sup> *<sup>z</sup>* (*n*) 0 1 *eiϑ* (155) = ⎛ ⎝ cosh(|*εMz* (*n*)|) <sup>−</sup> *<sup>ε</sup>Mz* (*n*) <sup>|</sup>*εMz* (*n*)<sup>|</sup> sinh(|*εMz* (*n*)|) <sup>−</sup> *<sup>ε</sup>M*<sup>∗</sup> *<sup>z</sup>* (*n*) <sup>|</sup>*εMz* (*n*)<sup>|</sup> sinh(|*εMz* (*n*)<sup>|</sup> cosh(|*εMz* (*n*)|) ⎞ ⎠ 1 *eiϑ* = cosh(|*εMz* (*n*)|) <sup>−</sup>*e*−*i*Φ*<sup>z</sup>* (*n*) sinh(|*εMz* (*n*)|) <sup>−</sup>*ei*Φ*<sup>z</sup>* (*n*) sinh(|*εMz* (*n*)<sup>|</sup> cosh(|*εMz* (*n*)|) 1 *eiϑ* ⎛ ⎜⎜⎝ *e i* <sup>2</sup> (*ϑ*−Φ*<sup>z</sup>* (*n*)) *e*−|*εMz* (*n*)<sup>|</sup> cos( <sup>1</sup> <sup>2</sup> (*<sup>ϑ</sup>* <sup>−</sup> <sup>Φ</sup>*<sup>z</sup>* (*n*)) <sup>−</sup> *ie*|*εMz* (*n*)<sup>|</sup> sin( <sup>1</sup> <sup>2</sup> (*<sup>ϑ</sup>* <sup>−</sup> <sup>Φ</sup>*<sup>z</sup>* (*n*)) *e i* <sup>2</sup> (*ϑ*+Φ*<sup>z</sup>* (*n*)) *e*−|*εMz* (*n*)<sup>|</sup> cos( <sup>1</sup> <sup>2</sup> (*<sup>ϑ</sup>* <sup>−</sup> <sup>Φ</sup>*<sup>z</sup>* (*n*)) + *ie*|*εMz* (*n*)<sup>|</sup> sin( <sup>1</sup> <sup>2</sup> (*<sup>ϑ</sup>* <sup>−</sup> <sup>Φ</sup>*<sup>z</sup>* (*n*)) ⎞ ⎟⎟⎠ (*n* ≥ 2),

where Φ*z*(*n*) is the phase

$$e^{-i\Phi\_{\overline{z}}(n)} = \frac{M\_{\overline{z}}(n)}{|M\_{\overline{z}}(n)|}. \tag{156}$$

From (155) follows an estimate of the norm (145)

$$||\chi||\_c^2 \simeq \mathcal{N}\_z(\varepsilon) \equiv \sum\_{n=1}^{\infty} \frac{1}{n} \left[ \varepsilon^{-2|cM\_z(n)|} (1 + \cos(\vartheta - \Phi\_z(n)) + \varepsilon^{2|cM\_z(n)|} (1 - \cos(\vartheta - \Phi\_z(n))) \right],\tag{157}$$

whose convergence depends on the asymptotic behavior of *Mz*(*n*) and Φ*z*(*n*). N*z*(*ε*) has the lower bound

$$\mathcal{N}\_z(\varepsilon) \ge \sum\_{n=1}^{\infty} \frac{2}{n} e^{-2|\varepsilon M\_z(n)|},\tag{158}$$

that follows from the inequality

$$a(1+b) + \frac{1}{a}(1-b) \ge 2a, \qquad a \in (0,1], \quad b \in [-1,1]. \tag{159}$$

If |*Mz*(*n*)| is bounded then the norm is infinite,

$$\text{if } |M\_{\overline{z}}(n)| < \mathbb{C}, \ \forall n \implies \mathcal{N}\_{\overline{z}}(\varepsilon) \ge \sum\_{n=1}^{\infty} \frac{2}{n} \varepsilon^{-2|\varepsilon|\mathbb{C}} = \infty. \tag{160}$$

This case corresponds in general to eigenstates belonging to the continuum. Eigenstates with finite norm require |*Mz*(*n*)| to be unbounded. Notice that N*z*(*ε*) is the sum of two series with non-negative terms. The convergence of the first summand in (157) is guaranteed if

$$\sum\_{n=1}^{\infty} \frac{1}{n} e^{-2|\varepsilon M\_{\tilde{z}}(n)|} < \infty \,\,\,\,\,\tag{161}$$

which occurs if |*Mz*(*n*)| diverges sufficiently fast with *n*. The convergence of the second summand in (157) requires Φ*z*(*n*) to have a limit when *n* → ∞, and to choose the parameter *ϑ* such that

$$\lim\_{n \to \infty} \Phi\_z(n) = \vartheta. \tag{162}$$

Moreover, 1 <sup>−</sup> cos(*<sup>ϑ</sup>* <sup>−</sup> <sup>Φ</sup>*z*(*n*)) must approach 0 sufficiently fast in order to compensate the factor <sup>1</sup> *<sup>n</sup> <sup>e</sup>*2*ε*|*Mz* (*n*)<sup>|</sup> . We now pass to analyze the latter conditions in detail.

## *12.3. Perron Formula*

Let us define the function

$$M\_z'(\mathbf{x}) \equiv \sum\_{1 \le n \le \mathbf{x}} \prime \frac{\mu(n)}{n^z}, \qquad z = \frac{1}{2} + iE, \quad E \in \mathbb{R}, \tag{163}$$

where ∑ <sup>1</sup>≤*n*≤*<sup>x</sup>* means that the last term in the sum is multiplied by 1/2 when *<sup>x</sup>* is an integer. Figure <sup>14</sup> shows |*M <sup>z</sup>*(*n*)| as a function of *E* for several values of *n*. Observe that |*M <sup>z</sup>*(*n*)| increases with *n* when *E* is a *zero*. We shall derive below this behavior.

**Figure 14.** Plot of |*M <sup>z</sup>*(*n*)| defined in Equation (163), for *E* ∈ (10, 23) and *n* = 50, 100, 150 (blue, orange, red curves) and 1/|*ζ*(1/2 + *iE*)| (black dotted line). Observe the increase with *n* at *E* = 14.13 ... and *E* = 21.02 . . . which are the first two *zeros* of *ζ*.

To compute *M <sup>z</sup>*(*x*) we use Perron's formula [76]

$$M\_z'(\mathbf{x}) = \lim\_{T \to \infty} \int\_{c-iT}^{c+iT} \frac{ds}{2\pi i} \frac{1}{\zeta(s+z)} \frac{\mathbf{x}^s}{\mathbf{s}}, \qquad c > \frac{1}{2} \,\tag{164}$$

where we have used that Re *z* = 1/2. The integral (164) can be done by residue calculus [42]

$$\lim\_{T \to \infty} \int\_{c - iT}^{c + iT} \frac{ds}{2\pi i} F(s) = \sum\_{\text{Res}\_{\boldsymbol{\gamma}} < \boldsymbol{c}} \text{Res}\_{\boldsymbol{s}\_{\boldsymbol{\gamma}}} F(s), \quad F(s) = \frac{1}{\zeta(s + z)} \frac{\mathbf{x}^s}{\mathbf{s}},\tag{165}$$

where the sum runs over the poles *sj* of *F*(*s*) located to the left of the line of integration Re *s* = *c*, which is Re *sj* < *c*. The poles of *F*(*s*) come from the zeros of *sζ*(*s* + *z*). The pole at *s* = 0 can be simple, or multiple, depending on the values of *ζ*(*z*) and its derivatives. The remaining poles of *F*(*s*) come from the zeros of *ζ*(*s* + *z*), say *sj* + *z* = *ρj*, and they lie to the left of the integration line, because the trivial and non-trivial *zeros* of *ζ*, satisfy Re *ρ<sup>j</sup>* < 1, which is

$$\operatorname{Re} s\_{\bar{j}} = \operatorname{Re} (\rho\_{\bar{j}} - z) = \operatorname{Re} \rho\_{\bar{j}} - \frac{1}{2} < \frac{1}{2} < c. \tag{166}$$

To compute the residues of Equation (165) we consider the cases: *s* = 0, *sj* + *z* a trivial zero of *ζ* and *sj* + *z* a non-trivial zero of *ζ*:

• *<sup>s</sup>* <sup>=</sup> 0. Let *<sup>m</sup>* <sup>≥</sup> 0 be the lowest integer such that *<sup>ζ</sup>*(*m*)(*z*) = *<sup>d</sup>mζ*(*z*)/*dz<sup>m</sup>* <sup>=</sup> 0. Then *<sup>F</sup>*(*s*) has a pole of order *m* + 1 at *s* = 0 with residue (The expression for Re*s*=<sup>0</sup> *F*(*s*) corresponding to the case *<sup>m</sup>* <sup>=</sup> 1 contains the term <sup>−</sup><sup>1</sup> <sup>2</sup> *ζ* (*z*)/(*ζ* (*z*))<sup>2</sup> which was omitted in the reference [42].)

$$\operatorname{Res}\_{s=0} F(s) = \begin{cases} 1/\zeta(z) & \text{if } \zeta(z) \neq 0, \\ \log x / \zeta'(z) - \frac{1}{2} \zeta''(z) / (\zeta'(z))^2 & \text{if } \zeta(z) = 0, \zeta'(z) \neq 0. \\ \vdots & \vdots \\ (\log x)^m / \zeta^{(m)}(z) + O((\log x)^{m-1}) & \text{if } \zeta(z) = \dots = \zeta^{(m-1)}(z) = 0, \zeta^{(m)}(z) \neq 0. \end{cases} \tag{167}$$

• *sn* = −2*n* − *z* (*n* = 1, 2, . . .), where *F*(*s*) has a simple pole due to the trivial *zeros* −2*n* of *ζ*.

$$\operatorname{Res}\_{s=-2n-z} F(s) = \frac{x^{-2n-z}}{-(2n+z)\zeta'(-2n)}, \qquad n = 1, 2, \ldots, \infty. \tag{168}$$

• *sj* = *ρ<sup>j</sup>* − *z* = 0, then *F*(*s*) has a pole due to the non-trivial zero *ρ<sup>j</sup>* of *ζ*

$$\text{Res}\_{\mathbf{s}=s\_{\hat{\boldsymbol{\rho}}}} F(\mathbf{s}) = \begin{cases} \frac{\mathbf{z}^{\boldsymbol{\rho}\_{\hat{\boldsymbol{\rho}}}^{\boldsymbol{\rho}\_{\hat{\boldsymbol{\rho}}}^{\boldsymbol{\rho}\_{\hat{\boldsymbol{\rho}}}^{\boldsymbol{\rho}}}}{(\boldsymbol{\rho}\_{\hat{\boldsymbol{\rho}}}-\mathbf{z})\zeta^{\mathbf{n}}(\boldsymbol{\rho}\_{\hat{\boldsymbol{\rho}}})}, & \text{if } \zeta(\boldsymbol{\rho}\_{\hat{\boldsymbol{\rho}}}) = 0, \zeta^{\mathbf{t}}(\boldsymbol{\rho}\_{\hat{\boldsymbol{\rho}}}) \neq 0 \\\\ \frac{m(\ln x)^{m-1}x^{\boldsymbol{\rho}\_{\hat{\boldsymbol{\rho}}}^{\boldsymbol{\rho}}}}{(\boldsymbol{\rho}\_{\hat{\boldsymbol{\rho}}}-\mathbf{z})\zeta^{\mathbf{n}}(\boldsymbol{\rho}\_{\hat{\boldsymbol{\rho}}})} + O((\ln x)^{m-2}), & \text{if } \zeta(\boldsymbol{\rho}\_{\hat{\boldsymbol{\rho}}}) = \dots = \zeta^{\mathbf{(m-1)}}(\boldsymbol{\rho}\_{\hat{\boldsymbol{\rho}}}) = 0, \zeta^{\mathbf{t}\mathbf{n}}(\boldsymbol{\rho}\_{\hat{\boldsymbol{\rho}}}) \neq 0, \; m \ge 2 \end{cases} \tag{169}$$

To make further progress we shall assume that all the Riemann zeros are simple, a statement which is not known to hold. The eventual case where there is a *zero* with double multiplicity will be considered elsewhere. In the former situation we are led to consider only two cases depending on whether *z* is, or is not, a simple *zero* of *ζ*. Collecting terms, we get

$$M\_{\mathbb{Z}}(\mathbf{x}) \quad = \frac{1}{\mathbb{Z}(z)} + \sum\_{\rho\_{\hat{\jmath}}} \frac{\mathbf{x}^{\rho\_{\hat{\jmath}} - z}}{(\rho\_{\hat{\jmath}} - z)\zeta'(\rho\_{\hat{\jmath}})} + \sum\_{n=1}^{\infty} \frac{\mathbf{x}^{-2n-z}}{-(2n+z)\zeta'(-2n)}, \quad \text{if } \zeta(z) \neq 0,\tag{170}$$

$$M\_{\overline{z}}(\mathbf{x}) \quad = \begin{array}{c} \frac{\log \underline{x}}{\overline{\zeta}'(\overline{z})} - \frac{\overline{\zeta}''(\overline{z})}{2(\overline{\zeta}'(\overline{z}))^2} + \sum\_{\overline{\rho}\_{\overline{\rho}} \neq \overline{\tau}} \frac{\overline{\zeta}^{\rho\_{\overline{\rho}}^{\circ} - \overline{z}}}{(\overline{\rho}\_{\overline{\tau}} - \overline{z})\overline{\zeta}'(\rho\_{\overline{\rho}})} + \sum\_{\overline{n}=1}^{\infty} \frac{\chi^{-2n-z}}{-(2n+z)\overline{\zeta}'(-2n)}, \quad \text{if } \overline{\zeta}(z) = 0, \zeta'(z) \neq 0. \end{array} \tag{171}$$

where the sum ∑*ρ<sup>j</sup>* runs over the non-trivial zeros of *ζ*. These equations are verified numerically in Figure 15. The last term in these equations, which comes from the trivial *zeros*, converges quickly and is finite for all *x* due to the exponential increase of *ζ* (−2*n*) [77]

$$\zeta'(-2n) = \frac{(-1)^n \zeta(2n+1)(2n)!}{2^{2n+1}\pi^{2n}} \stackrel{n \to \infty}{\longrightarrow} (-1)^n \sqrt{\pi n} \left(\frac{n}{e\pi}\right)^{2n}.\tag{172}$$

**Figure 15.** Plot of |*M <sup>z</sup>*(*n*)| for *n* = 10, ... , 50 and *E* = 20 (**left**) and *E* = 14.13 (**right**). In red the values obtained doing the sum in Equation (163). In blue the sum of Equation (170) for *E* = 20 and Equation (171) for *E* = 14.13, including the first 100 Riemann zeros, and 20 trivial zeros. Observe the accuracy of the approximation. The slow increase in the latter plot is due to the factor log *n* in Equation (171).

We do not know an estimation of the term depending on the sum over the non-trivial zeros. If the Riemann hypothesis is true the term *<sup>x</sup>ρj*−*<sup>z</sup>* will oscillate as a function of *<sup>x</sup>*. We expect that for *<sup>ζ</sup>*(*z*) <sup>=</sup> 0, |*Mz*(*x*)| will not yield a finite norm such that the corresponding eigenstate will not belong to the discrete spectrum. When *ζ*(*z*)) = 0, *ζ* (*z*) = 0, we shall make the approximation

$$M\_z(\mathbf{x}) \quad \rightarrow \quad \frac{\log \mathbf{x}}{\zeta'(z)} \qquad \mathbf{x} \rightarrow \infty \,, \tag{173}$$

where we neglect the finite part *<sup>ζ</sup>* (*z*) <sup>2</sup>(*<sup>ζ</sup>* (*z*))<sup>2</sup> ; and the possible contribution of the sum over the Riemann zeros. Using that *ζ*(1/2 + *iE*) = *e*−*iθ*(*E*)*Z*(*E*) we find

$$M\_{\mathbf{x}}(z) \quad \rightarrow \quad \frac{i\,\epsilon^{i\theta(E)}\log\mathbf{x}}{Z'(E)} \quad \text{as} \quad \mathbf{x} \rightarrow \infty,\tag{174}$$

hence Φ*z*(*n*), given in Equation (156), behaves as

$$e^{-i\Phi\_z(n)} \to i e^{i\theta(E)} \text{sign} \, Z'(E) \quad \text{as} \quad n \to \infty \,, \tag{175}$$

which has a well-defined asymptotic limit. We shall then choose *ϑ* according to Equation (162) namely

$$\theta = \lim\_{n \to \infty} \Phi\_{\overline{z}}(n) = -\left(\theta(E) + \frac{\pi}{2} \text{sign} \, Z'(E)\right),\tag{176}$$

that provides a necessary condition for the convergence of the norm. It remains to show that Equation (176) is also sufficient but this requires the knowledge of the next to leading correction to (174). Notice that *ϑ* depends on *θ*(*E*) and the sign of *Z* (*E*), a feature that is not left fixed in Equation (142). The norm (157) then becomes

$$||\chi||\_{\varepsilon}^{2} \simeq \sum\_{n=1}^{\infty} \frac{2}{n} e^{-2\varepsilon \log n / |Z'(E)|} = 2\tilde{\zeta} \left( 1 + \frac{2\varepsilon}{|Z'(E)|} \right) < \infty,\tag{177}$$

that is finite for all *ε* > 0. This result indicates that a *zero* of the zeta function gives a normalizable state, in agreement with heuristic derivation proposed in the previous section, but there are some differences. First, the eigenvalue *E* does not need to be expanded in series of *ε*. It is taken to be a *zero* of *ζ* from the beginning. This choice generates the log *x* term in Equation (171) and is responsible for the finiteness of the norm after the appropriate choice of the phase (176) that also differs from the heuristic value (142). On the other hand, if *ϑ* does not satisfy Equation (176), then the norm of the state will diverge badly and so the *zero E* will be missing in the spectrum. Finally, if *E* is not a *zero*, we expect that the state will belong generically to the continuum. Figure 16 shows the expected spectrum of the model, which recalls Connes's scenario of missing spectral lines, except that in our case, one can pick up a zero at a time by tuning *ϑ*.

**Figure 16.** Graphical representation of the spectrum of the model. It is expected to consist of an infinite number of bands separated by forbidden regions of width proportional to *ε*. The latter regions may contain a *zero En* if the phase *ϑ* is chosen according to Equation (176). Otherwise, the *zeros* will be missing in the spectrum that is represented by the points *En*−<sup>1</sup> and *En*+1.

If the RH is false there will be at least four *zeros* outside the critical line, say *ρ<sup>c</sup>* = *σ<sup>c</sup>* + *iEc*, *ρ*¯*<sup>c</sup>* = *σ<sup>c</sup>* − *iEc*, 1 <sup>−</sup> *<sup>ρ</sup><sup>c</sup>* and 1 <sup>−</sup> *<sup>ρ</sup>*¯*c*, with *<sup>σ</sup><sup>c</sup>* <sup>&</sup>gt; <sup>1</sup> <sup>2</sup> , *Ec* ∈ R+. We shall choose the highest value of *σc*. The asymptotic behavior of *Mz*(*x*) will be given by the *zeros* located to the right of the critical line,

$$M\_{\tilde{z}}(x) \quad \rightarrow \quad \frac{x^{\rho\_{\varepsilon}-z}}{(\rho\_{\varepsilon}-z)\zeta'(\rho\_{\varepsilon})} + \frac{x^{\rho\_{\varepsilon}-z}}{(\overline{\rho\_{\varepsilon}-z})\zeta'(\overline{\rho\_{\varepsilon}})} \quad \text{as} \; \ge \; \sim \; \tag{178}$$

To simplify the discussion let us choose *E Ec*, which yields the approximation

$$M\_{\mathbf{z}}(\mathbf{x}) \to \frac{2i \, \mathbf{x}^{\sigma\_{\mathbf{c}} - 1/2 - i \mathbf{E}}}{E|\zeta'(\rho\_{\mathbf{c}})|} \cos(E\_{\mathbf{c}} \log \mathbf{x} - \phi\_{\mathbf{c}}) \quad \text{as} \ \mathbf{x} \to \infty,\tag{179}$$

where *eiφ<sup>c</sup>* = *ζ* (*ρc*)/|*ζ* (*ρc*)|. The phase Φ*z*(*n*) is given by Equation (156)

$$\Phi\_{\mathbf{z}}(n) \to E \log n - \frac{\pi}{2} \text{sign} \left( \cos(E\_{\mathbf{c}} \log n - \phi\_{\mathbf{c}}) \right) \quad \text{as} \quad n \to \infty. \tag{180}$$

Correspondingly, the norm (157) diverges so badly, ∝ ∑*<sup>n</sup>* <sup>1</sup> *<sup>n</sup>* exp(*Cnσc*−1/2)... , for any value of *ϑ* that the state will not be normalizable even using Dirac delta functions. This result occurs for all eigenenergies *E*. Therefore, the Hamiltonian will not admit a spectral decomposition, but this is impossible because it is a well-defined self-adjoint operator. We conclude that a *zero* outside the critical line does not exist which provides an argument likely to be persuasive to physicists for the truth of the Riemann hypothesis.

## **13. The Riemann Interferometer**

The model considered in the previous sections looks at first glance quite difficult to simulate. We shall next show that this model is equivalent to another one that can be implemented in the Lab. We shall call this system the Riemann interferometer. The basic idea can be illustrated with the mapping between the quantum *xp* Hamiltonian and the momentum operator *p*ˆ. Let us make the change of coordinates *x* = log *ρ* and relate the wave functions in both coordinates, *φ*(*x*) and *ψ*(*ρ*), as follows

$$
\phi(\mathbf{x}) = \left(\frac{d\rho}{d\mathbf{x}}\right)^{1/2} \psi(\rho) = \epsilon^{\mathbf{x}/2} \psi(\epsilon^{\mathbf{x}}) \,. \tag{181}
$$

An eigenstate of the Hamiltonian (*ρ p*ˆ*<sup>ρ</sup>* + *p*ˆ*<sup>ρ</sup> ρ*)/2, with eigenvalue *E*, is mapped by Equation (181) into an eigenstate of the momentum operator *p*ˆ*<sup>x</sup>* = −*i∂<sup>x</sup>* with the same eigenvalue,

$$\psi(\rho) = \frac{1}{\rho^{1/2 - iE}} \implies \phi(\mathbf{x}) = e^{iEx}. \tag{182}$$

*Symmetry* **2019**, *11*, 494

This shows that the energy *E* can be seen as momentum. For a relativistic massless fermion, this is always the case. The measure that defines the scalar product of the corresponding Hilbert spaces are one-to-one related

$$\int\_{\ell}^{\infty} d\rho \,\psi\_1^\*(\rho)\psi\_2(\rho) = \int\_{\log \ell}^{\infty} d\mathbf{x} \,\phi\_1^\*(\mathbf{x})\phi\_2(\mathbf{x})\,. \tag{183}$$

The operator (*ρ p*ˆ*<sup>ρ</sup>* + *p*ˆ*<sup>ρ</sup> ρ*)/2 is self-adjoint in the interval (0, ∞) but not in the interval (1, ∞), just like *p*ˆ*<sup>x</sup>* is self-adjoint in the real line (−∞, ∞) but not in the half-line (0, ∞) [23,66]. The former case corresponds to the value - = 0 and the latter one to - = 1 in Equation (183). Let us now consider the Dirac Hamiltonian in the Rindler variable *ρ*, given in Equation (115). It becomes in the *x* variable

$$H = \begin{pmatrix} -i\partial\_x & 0\\ 0 & i\partial\_x \end{pmatrix} \tag{184}$$

Unlike *p*ˆ*x*, this Hamiltonian is self-adjoint in the interval *x* ∈ (log -1, ∞). We choose for convenience -<sup>1</sup> = 1. The moving mirrors located at *ρ* = *<sup>n</sup>* are now placed at the positions *x* = *dn*, with *dn* = log *<sup>n</sup>*, so for *<sup>n</sup>* <sup>=</sup> <sup>√</sup>*n*, we get

$$d\_n = \frac{1}{2} \log n\,,\tag{185}$$

where *n* are square free integers and the reflection coefficients are given by *rn* = *μ*(*n*)/ <sup>√</sup>*n*. Figure <sup>17</sup> shows the array of mirrors satisfying Equation (185). One can easily generalize this interferometer to provide a spectral realization of the *zeros* of Dirichlet *L*-functions, by changing the reflection coefficients *rn*,

$$L\_{\chi}(s) = \sum\_{n=1}^{\infty} \frac{\chi(n)}{n^s} \longrightarrow r\_n = \frac{\mu(n)\,\chi(n)}{n^{1/2}},\tag{186}$$

where *χ*(*n*) is the Dirichlet character associated with the *L*-function. It would be interesting to replace the massless fermions by massless bosons, say photons and study what kind of Riemann interferometer arise.

**Figure 17.** Graphical representation of the array of mirrors in Minkowski space that reproduce the Riemann zeros. The phase at the boundary *ϑ* must be chosen according to Equation (176) in order that *E* is an eigenvalue of the Hamiltonian. Recall Figure 13. Between the mirrors the wave functions are plane waves.

## **14. Dirac Models for a Class of Modified** *ζ* **and** *L* **Functions**

Grosswald and Schnitzer proved in 1978 two very surprising theorems that we shall use below to generalize the construction done in the previous sections. Let us first consider a set on integers *qn* satisfying the conditions

$$p\_n \le q\_n \le p\_{n+1}, \qquad n = 1, \dots, \infty,\tag{187}$$

where *pn* is the *n*th prime number. With these numbers define the infinite product

$$\zeta^\*(s) = \prod\_{n=1}^{\infty} (1 - q\_n^{-s})^{-1}. \tag{188}$$

One then has [78]:

**Theorem 1.** *This function is holomorphic for σ* = Re *s* > 1 *and has the following properties:*


This theorem means that the relation between prime numbers and Riemann zeros via the zeta function is less rigid that one may have though. We shall use this freedom to associate a Hamiltonian to every series satisfying (187). Let us first write the inverse of (188) as

$$\frac{1}{\zeta^\*(s)} = \sum\_{n=1}^{\infty} \frac{\mu^\*(n)}{n^s}, \qquad \mu^\*(n) = n\_{\text{even}} - n\_{\text{odd}}.\tag{189}$$

where *n*even(*n*odd) is the number of times *n* can be written as the product of an even (odd) number of *qi* numbers in the series (187). An example of a series satisfying (187) is

$$2, 4, 6, 8, 12, \dots, q\_n = p\_n + 1, \dots \tag{190}$$

for which we have

$$\frac{1}{\zeta^\*(s)} = 1 - \frac{1}{2^s} - \frac{2}{(2^6 \cdot 3)^s} + \frac{2}{(2^3 \cdot 3)^s} + \frac{1}{(2^8 \cdot 3)^s} - \frac{1}{2^{2s}} - \frac{1}{(2 \cdot 3)^s} + \dots \tag{191}$$

Notice that *<sup>μ</sup>*∗(2<sup>6</sup> · <sup>3</sup>) = −2 because 2<sup>6</sup> · <sup>3</sup> = <sup>4</sup> · <sup>6</sup> · <sup>8</sup> = <sup>2</sup> · <sup>8</sup> · 12. Obviously *<sup>μ</sup>*∗(*n*) = *<sup>μ</sup>*(*n*) if *qn* = *pn*, ∀*n*. Using Equation (189) we define a massless Dirac model with reflection coefficients (recall Equation (153))

$$r\_n = \frac{\mu^\*(n)}{n^{1/2}}, \quad n > 1. \tag{192}$$

Hence, by the arguments given in Section 12 and theorem 1, we shall find the Riemann zeros in the spectrum of the Hamiltonian *H<sup>ϑ</sup>* by tuning the parameter *ϑ* in the limit *ε* → 0.

The second theorem in reference [78] is an extension of theorem 1 to Dirichlet *L*-functions *L*(*s*) = <sup>∏</sup>*n*(<sup>1</sup> − *<sup>χ</sup>*(*n*)*n*−*s*)−1, where *<sup>χ</sup>* is a character modulo *<sup>k</sup>*. The series (187) is replaced by

$$p\_n \le q\_n \le p\_n + \mathcal{K}\_\prime \qquad p\_n = q\_n \bmod k \tag{193}$$

where *K* ≥ *k*. The modified *L*-function is defined as

$$L^\*(s) = \prod\_{n=1}^{\infty} (1 - \chi(q\_n) q\_n^{-s})^{-1},\tag{194}$$

that can be extended to the region *σ* > 0, with the same zeros (and multiplicities) as *L*(*s*). In this case, too, we can construct a Dirac model with reflection coefficients (recall Equation (186))

$$r\_{\rm ll} = \frac{\chi(n)\mu^\*(n)}{n^{1/2}}, \quad n > 1. \tag{195}$$

whose associated Hamiltonian *H<sup>ϑ</sup>* contains the zeros of *L*(*s*) by varying *ϑ*. Theorem 2 of [78] was mentioned by LeClair and Mussardo in [63] as a support to their approach to the Generalized Riemann hypothesis based on random walks and the Lemke Oliver-Soundararajan conjecture on the distribution of pairs of residues on consecutive primes [79] (for other statistical properties of the prime numbers see [80,81]). It will be worth to investigate if there is a relation between our approach and the one proposed in [62,63].

## **15. Conclusions**

In this paper, we have reviewed the spectral approach to the RH that started with the Berry–Keating–Connes *xp* model and continued with several works aimed to provide a physical realization of the Riemann zeros. The main steps in this approach are: (i) spectral realization of Connes's *xp* model using the Landau model of an electron in a magnetic field and electrostatic potential, (ii) construction of modified quantum *xp* models whose spectra reproduce, on average, the behavior of the *zeros*, (iii) reformulation of the *x*(*p* + 1/*p*) model as a relativistic theory of a massive Dirac fermion in a region of Rindler spacetime, (iv) inclusion of prime numbers into the massless Dirac equation by means of delta function potentials acting as moving mirrors that, in the limit where they become semitransparent, leads to a spectral realization of the *zeros*, (v) a route for proving the Riemann Hypothesis, and (vi) proposal of an interferometer that may provide an experimental observation of the zeros of the Riemann zeta function and other Dirichlet *L*-functions.

The Pólya-Hilbert (PH) conjecture was proposed as a physical explanation of the RH based on the spectral properties of self-adjoint operators: there exists a *single* quantum Hamiltonian containing *all* the Riemann zeros in its spectrum which are therefore real numbers. This statement can be called the *global* version of the PH conjecture. Instead of this, we have found a *local* version according to which a Riemann zero *En* becomes an eigenvalue of the Hamiltonian *H<sup>ϑ</sup>* provided the parameter *ϑ*, which characterizes the self-adjoint extension, is fine-tuned to the combination *θ*(*En*) + *<sup>π</sup>* <sup>2</sup> sign*Z* (*En*). In this sense the Hamiltonian provides a physical realization of *ζ*( <sup>1</sup> <sup>2</sup> + *it*), and not only of the Riemann-Siegel *Z* function. We have given arguments for a proof of the RH by contradiction: the existence of a *zero* off the critical line implies that the eigenstates of *H<sup>ϑ</sup>* are non-normalizable in the discrete or continuum sense, which is impossible since *H<sup>ϑ</sup>* is a self-adjoint operator. These results are obtained in the limit where the mirrors become semitransparent and assumes the convergence of some mathematical series that need to be analyzed more thoroughly. Finally, we have proposed an interferometer made of fermions propagating in an array of mirrors that may yield an experimental observation of the Riemann zeros in the Lab.

**Funding:** Grants FIS2012-33642, FIS2015-69167-C2-1-P, QUITEMAD+ S2013/ICE-2801; and SEV-2012-0249, and SEV-2016-0597 of the "Centro de Excelencia Severo Ochoa" Program.

**Acknowledgments:** I am grateful for fruitful discussions and comments to Julio Andrade, Manuel Asorey, Michael Berry, Ignacio Cirac, Charles Creffield, Jon Keating, José Ignacio Latorre, Giuseppe Mussardo, André LeClair, Miguel Angel Martín-Delgado, Javier Molina-Vilaplana, Javier Rodríguez-Laguna, Mark Srednicki and Paul Townsend. I thank Denis Bernard for pointing out an error in the first version of this manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Universal Quantum Computing and Three-Manifolds**

## **Michel Planat 1,\*, Raymond Aschheim 2, Marcelo M. Amaral <sup>2</sup> and Klee Irwin <sup>2</sup>**


Received: 23 November 2018; Accepted: 14 December 2018; Published: 19 December 2018

**Abstract:** A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere *S*3. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold *M*3. More precisely, the *d*-dimensional POVMs defined from subgroups of finite index of the modular group *PSL*(2,Z) correspond to *d*-fold *M*3- coverings over the trefoil knot. In this paper, we also investigate quantum information on a few 'universal' knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and *M*3's obtained from Dehn fillings are explored.

**Keywords:** quantum computation; IC-POVMs; knot theory; three-manifolds; branch coverings; Dehn surgeries

**PACS:** 03.67.Lx; 03.65.Wj; 03.65.Aa; 02.20.-a; 02.10.Kn; 02.40.Pc; 02.40.Sf

**MSC:** 81P68; 81P50; 57M25; 57R65; 14H30; 20E05; 57M12

*Manifolds are around us in many guises.*

*As observers in a three-dimensional world, we are most familiar with two-manifolds: the surface of a ball or a doughnut or a pretzel, the surface of a house or a tree or a volleyball net...*

*Three-manifolds may be harder to understand at first. But as actors and movers in a three-dimensional world, we can learn to imagine them as alternate universes.*

(William Thurston [1]).

## **1. Introduction**

Mathematical concepts pave the way for improvements in technology. As far as topological quantum computation is concerned, non-abelian anyons have been proposed as an attractive (fault-tolerant) alternative to standard quantum computing which is based on a universal set of quantum gates [2–5]. Anyons are two-dimensional quasiparticles with world lines forming braids in space-time. Whether non-abelian anyons do exist in the real world and/or would be easy to create artificially, is still open to discussion. In this paper, we propose an alternative to anyon-based universal quantum computation (UQC) thanks to three-dimensional topology. Our proposal relies on appropriate 3-manifolds whose fundamental group is used for building the magic states for UQC. Three-dimensional topological quantum computing would federate the foundations of quantum mechanics and cosmology, a recurrent dream of many physicists. Three-dimensional topology was

already investigated by several groups in the context of quantum information [6,7], high energy physics [8,9], biology [10] and consciousness studies [11].

Recall the context of our work and clarify its motivation. Bravyi & Kitaev introduced the principle of 'magic state distillation': universal quantum computation, the possibility to implement an arbitrary quantum gate, may be realized thanks to the stabilizer formalism (Clifford group unitaries, preparations and measurements) and the ability to prepare an appropriate single qubit non-stabilizer state, called a 'magic state' [12]. Then, irrespectively of the dimension of the Hilbert space where the quantum states live, a non-stabilizer pure state was called a magic state [13]. An improvement of this concept was carried out in [14,15] showing that a magic state could be at the same time a fiducial state for the construction of an informationally complete positive operator-valued measure, or IC-POVM, under the action on it of the Pauli group of the corresponding dimension. Thus UQC in this view happens to be relevant both to such magic states and to IC-POVMs. In [14,15], a *d*-dimensional magic state follows from the permutation group that organizes the cosets of a subgroup *H* of index *d* of a two-generator free group *G*. This is due to the fact that a permutation may be seen as a permutation matrix/gate and that mutually commuting matrices share eigenstates—they are either of the stabilizer type (as elements of the Pauli group) or of the magic type. In the calculation, it is enough to keep magic states that are simultaneously fiducial states for an IC-POVM and we are done. Remarkably, a rich catalog of the magic states relevant to UQC and IC-POVMs can be obtained by selecting *G* as the two-letter representation of the modular group Γ = *PSL*(2,Z) [16]. The next step, developed in this paper, is to relate the choice of the starting group *G* to three-dimensional topology. More precisely, *G* is taken as the fundamental group *<sup>π</sup>*1(*S*<sup>3</sup> \ *<sup>K</sup>*) of a 3-manifold *<sup>M</sup>*<sup>3</sup> defined as the complement of a knot or link *K* in the 3-sphere *S*3. A branched covering of degree *d* over the selected *M*<sup>3</sup> has a fundamental group corresponding to a subgroup of index *d* of *π*<sup>1</sup> and may be identified as a sub-manifold of *M*3, the one leading to an IC-POVM is a model of UQC. In the specific case of Γ, the knot involved is the left-handed trefoil knot *T*1, as shown in Section 2.

While Γ serves as a motivation for investigating the trefoil knot manifold in relation to UQC and the corresponding ICs, it is important to put the UQC problem in the wider frame of Poincaré conjecture, the Thurston's geometrization conjecture and the related 3-manifolds [1]. For example, ICs may also follow from hyperbolic or Seifert 3-manifolds as shown in Tables of this paper.

More details are provided at the next subsections.

#### *1.1. From Poincaré Conjecture to UQC*

The Poincaré conjecture is the elementary (but deep) statement that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere *S*<sup>3</sup> [17]. Having in mind the correspondence between *<sup>S</sup>*<sup>3</sup> and the Bloch sphere that houses the qubits *<sup>ψ</sup>* <sup>=</sup> *<sup>a</sup>* <sup>|</sup>0 <sup>+</sup> *<sup>b</sup>* <sup>|</sup>1, *<sup>a</sup>*, *<sup>b</sup>* <sup>∈</sup> <sup>C</sup>, <sup>|</sup>*a*<sup>|</sup> <sup>2</sup> + |*b*| <sup>2</sup> = 1, one would desire a quantum translation of this statement. For doing this, one may use the picture of the Riemann sphere C ∪ ∞ in parallel to that of the Bloch sphere and follow F. Klein lectures on the icosahedron to perceive the platonic solids within the landscape [18]. This picture fits well the Hopf fibrations [19], their entanglements described in [20,21] and quasicrystals [22,23]. However, we can be more ambitious and dress *S*<sup>3</sup> in an alternative way that reproduces the historic thread of the proof of Poincaré conjecture. Thurston's geometrization conjecture, from which Poincaré conjecture follows, dresses *S*<sup>3</sup> as a 3-manifold not homeomorphic to *S*3. The wardrobe of 3-manifolds *M*<sup>3</sup> is huge but almost every dress is hyperbolic and W. Thurston found the recipes for them [1]. Every dress is identified thanks to a signature in terms of invariants. For our purpose, the fundamental group *π*<sup>1</sup> of *M*<sup>3</sup> does the job.

The three-dimensional space surrounding a knot *<sup>K</sup>*—the knot complement *<sup>S</sup>*<sup>3</sup> \ *<sup>K</sup>*—is an example of a three-manifold [1,24]. We will be especially interested by the trefoil knot that underlies work of the first author [16] as well as the figure-of-eight knot, the Whitehead link and the Borromean rings because they are universal (in a sense described below), hyperbolic and allow to build 3-manifolds from platonic manifolds [25]. Such manifolds carry a quantum geometry corresponding to quantum computing and (possibly informationally complete) POVMs identified in our earlier work [14–16].

According to [26], the knot *<sup>K</sup>* and the fundamental group *<sup>G</sup>* = *<sup>π</sup>*1(*S*<sup>3</sup> \ *<sup>K</sup>*) are universal if every closed and oriented 3-manifold *M*<sup>3</sup> is homeomorphic to a quotient H/*G* of the hyperbolic 3-space H by a subgroup *H* of finite index *d* of *G*. As just announced, the figure-of-eight knot, the Whitehead link and Borromean rings are universal. The catalog of the finite index subgroups of their fundamental group *G* and of the corresponding 3-manifolds defined from the *d*-fold coverings [27] can easily be established up to degree 8, using the software SnapPy [28].

In paper [16] of the first author, it has been found that minimal *d*-dimensional IC-POVMs (sometimes called MICs) may be built from finite index subgroups of the modular group Γ = *PSL*(2,Z). To such an IC (or MIC) is associated a subgroup of index *d* of Γ, a fundamental domain in the Poincaré upper-half plane and a signature in terms of genus, elliptic points and cusps as summarized in ([16] Figure 1). There exists a relationship between the modular group Γ and the trefoil knot *T*<sup>1</sup> since the fundamental group *<sup>π</sup>*1(*S*<sup>3</sup> \ *<sup>T</sup>*1) of the knot complement is the braid group *<sup>B</sup>*3, the central extension of Γ. However, the trefoil knot and the corresponding braid group *B*<sup>3</sup> are not universal [29] which forbids the relation of the finite index subgroups of *B*<sup>3</sup> to all three-manifolds.

It is known that two coverings of a manifold *M* with fundamental group *G* = *π*1(*M*) are equivalent if there exists a homeomorphism between them. Besides, a *d*-fold covering is uniquely determined by a subgroup of index *d* of the group *G* and the inequivalent *d*-fold coverings of *M* correspond to conjugacy classes of subgroups of *G* [27]. In this paper we will fuse the concepts of a three-manifold *M*<sup>3</sup> attached to a subgroup *H* of index *d* and the POVM, possibly informationally complete (IC), found from *H* (thanks to the appropriate magic state and related Pauli group factory).

## *1.2. Minimal Informationally Complete POVMs and UQC*

In our approach [15,16], minimal informationally complete (IC) POVMs are derived from appropriate fiducial states under the action of the (generalized) Pauli group. The fiducial states also allow to perform universal quantum computation [14].

A POVM is a collection of positive semi-definite operators {*E*1, ... , *Em*} that sum to the identity. In the measurement of a state *ρ*, the *i*-th outcome is obtained with a probability given by the Born rule *p*(*i*) = tr(*ρEi*). For a minimal IC-POVM (or MIC), one needs *d*<sup>2</sup> one-dimensional projectors Π*<sup>i</sup>* = <sup>|</sup>*ψi ψi*|, with <sup>Π</sup>*<sup>i</sup>* <sup>=</sup> *dEi*, such that the rank of the Gram matrix with elements tr(Π*i*Π*j*), is precisely *<sup>d</sup>*2. A SIC-POVM (the *S* means symmetric) obeys the relation" " # *ψi*|*ψ<sup>j</sup>* \$" "<sup>2</sup> <sup>=</sup> tr(Π*i*Π*j*) = *<sup>d</sup>δij*+<sup>1</sup> *<sup>d</sup>*+<sup>1</sup> , that allows the explicit recovery of the density matrix as in ([30] Equation (29)).

New minimal IC-POVMs (i.e., whose rank of the Gram matrix is *d*2) and with Hermitian angles " " # *ψi*|*ψ<sup>j</sup>* \$" " *<sup>i</sup>*=*<sup>j</sup>* <sup>∈</sup> *<sup>A</sup>* <sup>=</sup> {*a*1, ... , *al*} have been discovered [16]. A SIC (i.e., a SIC-POVM) is equiangular with <sup>|</sup>*A*<sup>|</sup> <sup>=</sup> 1 and *<sup>a</sup>*<sup>1</sup> <sup>=</sup> <sup>√</sup> <sup>1</sup> *d*+1 . The states encountered are considered to live in a cyclotomic field F = Q[exp( <sup>2</sup>*i<sup>π</sup> <sup>n</sup>* )], with *n* = GCD(*d*,*r*), the greatest common divisor of *d* and *r*, for some *r*. The Hermitian angle is defined as " " # *ψi*|*ψ<sup>j</sup>* \$" " *<sup>i</sup>*=*<sup>j</sup>* <sup>=</sup> % %(*ψi*, *<sup>ψ</sup>j*) % % 1 deg , where . means the field norm of the pair (*ψi*, *<sup>ψ</sup>j*) in F and deg is the degree of the extension F over the rational field Q [15].

The fiducial states for SIC-POVMs are quite difficult to derive and seem to follow from algebraic number theory [31].

Except for *d* = 3, the IC-POVMs derived from permutation groups are not symmetric and most of them can be recovered thanks to subgroups of index *d* of the modular group Γ ([16] Table 2).The geometry of the qutrit Hesse SIC is shown in Figure 1a. It follows from the action of the qutrit Pauli group on magic/fiducial states of type (0, 1, ±1). For *d* = 4, the action of the two-qubit Pauli group on the magic/fiducial state of type (0, 1, <sup>−</sup>*ω*6, *<sup>ω</sup>*<sup>6</sup> <sup>−</sup> <sup>1</sup>) with *<sup>ω</sup>*<sup>6</sup> <sup>=</sup> exp( <sup>2</sup>*i<sup>π</sup>* 6 ) results into a minimal IC-POVM whose geometry of triple products of projectors Π*<sup>i</sup>* turns out to correspond to the commutation graph of Pauli operators, see Figure 1b and ([16] Figure 2). For *d* = 5, the geometry of an IC consists of copies of the Petersen graph reproduced in Figure 1c. For *d* = 6, the geometry consists of components looking like Borromean rings (see [16] Figure 2 and Table 1 below).

**Figure 1.** Geometrical structure of low dimensional MICs: (**a**) the qutrit Hesse SIC, (**b**) the two-qubit MIC that is the generalized quadrangle of order two *GQ*(2, 2), (**c**) the basic component of the 5-dit MIC that is the Petersen graph. The coordinates on each diagram are the *d*-dimensional Pauli operators that act on the fiducial state, as shown.

#### *1.3. Organization of the Paper*

The paper is organized as follows. Section 2 deals with the relationship between quantum information seen from the modular group Γ and from the trefoil knot 3-manifold. Section 3 deals with the (platonic) 3-manifolds related to coverings over the figure-of-eight knot, Whitehead link and Borromean rings, see Figure 2, and how they relate to minimal IC-POVMs. Section 4 describes the important role played by Dehn fillings for describing the many types of 3-manifolds that may relate to topological quantum computing.

**Figure 2.** (**a**) The figure-of-eight knot: *K*4*a*1 = otet0200001 = *m*004, (**b**) the Whitehead link *L*5*a*1 = ooct0100001 = *m*129, (**c**) Borromean rings *L*6*a*4 = ooct0200005 = *t*12067.

## **2. Quantum Information from the Modular Group** Γ **and the Related Trefoil Knot** *T*<sup>1</sup>

In this section, we describe the results established in [16] in terms of the 3-manifolds corresponding to coverings of the trefoil knot complement *<sup>S</sup>*<sup>3</sup> \ *<sup>T</sup>*1.

Let us introduce to the group representation of a knot complement *<sup>π</sup>*1(*S*<sup>3</sup> \ *<sup>K</sup>*). A Wirtinger representation is a finite representation of *π*<sup>1</sup> where the relations are of the form *wgiw*−<sup>1</sup> = *gj* where

*w* is a word in the *k* generators {*g*1, ··· , *gk*}. For the trefoil knot *T*<sup>1</sup> = *K*3*a*1 = 31 shown in Figure 3a, a Wirtinger representation is [32]

$$
\pi\_1(S^3 \backslash T\_1) = \langle \mathfrak{x}, y | y \mathfrak{x} y = \mathfrak{x} y \mathfrak{x} \rangle \text{ or equivalently } \quad \pi\_1 = \left\langle \mathfrak{x}, y | y^2 = \mathfrak{x}^3 \right\rangle.
$$

In the rest of the paper, the number of *d*-fold coverings of the manifold *M*<sup>3</sup> corresponding to the knot *T* will be displayed as the ordered list *ηd*(*T*), *d* ∈ {1..10 . . .}. For *T*<sup>1</sup> it is

$$\eta\_d(T\_1) = \{1, 1, 2, 3, 2, 8, 7, 10, 18, 28, \dots\} \dots$$

Details about the corresponding *d*-fold coverings are in Table 1. As expected, the coverings correspond to subgroups of index *d* of the fundamental group associated to the trefoil knot *T*1.

**Figure 3.** (**a**) The trefoil knot *T*<sup>1</sup> = *K*3*a*1 = 31, (**b**) the link *L*7*n*1 associated to the Hesse SIC, (**c**) the link *L*6*a*3 associated to the two-qubit IC.

#### *2.1. Cyclic Branched Coverings over the Trefoil Knot*

Let *p*, *q*,*r* be three positive integers (with *p* ≤ *q* ≤ *r*), the Brieskorn 3-manifold Σ(*p*, *q*,*r*) is the intersection in C<sup>3</sup> of the 5-sphere *S*<sup>5</sup> with the surface of equation *z p* <sup>1</sup> + *z q* <sup>2</sup> + *<sup>z</sup><sup>r</sup>* <sup>3</sup> = 1. In [33], it is shown that a *r*-fold cyclic covering over *S*<sup>3</sup> branched along a torus knot or link of type (*p*, *q*) is a Brieskorn 3-manifold Σ(*p*, *q*,*r*) (see also Section 4.1). For the spherical case *p*−<sup>1</sup> + *q*−<sup>1</sup> + *r*−<sup>1</sup> > 1, the group associated to a Brieskorn manifold is either dihedral [that is the group *Dr* for the triples (2, 2,*r*)], tetrahedral [that is *A*<sup>4</sup> for (2, 3, 3)], octahedral [that is *S*<sup>4</sup> for (2, 3, 4)] or icosahedral [that is *A*<sup>5</sup> for (2, 3, 5)]. The Euclidean case *p*−<sup>1</sup> + *q*−<sup>1</sup> + *r*−<sup>1</sup> = 1 corresponds to (2, 3, 6), (2, 4, 4) or (3, 3, 3). The remaining cases are hyperbolic.

The cyclic branched coverings with spherical groups for the trefoil knot (which is of type (2, 3)) are identified in the right hand side column of Table 1.

## *2.2. Irregular branched coverings over the trefoil knot*

The right hand side column of Table 1 shows the subgroups of Γ identified in ([16] Table 1) as corresponding to a minimal IC-POVM. Let us give a few more details on how to attach a MIC to some coverings/subgroups of the trefoil knot fundamental group *π*1(*T*1). Columns 1 to 6 in Table 1 contain information available in SnapPy [28], with *d*, ty, hom, cp, gens and CS the degree, the type, the first homology group, the number of cusps, the number of generators and the Chern-Simons invariant of the relevant covering, respectively. In column 7, a link is possibly identified by SnapPy when the fundamental group and other invariants attached to the covering correspond to those of the link. For our purpose, we are also interested in the possible recognition of a MIC behind some manifolds in the table.

**Table 1.** Coverings of degree *d* over the trefoil knot found from SnapPy [28]. The related subgroup of modular group Γ and the corresponding IC-POVM [16] (when applicable) is in the right column. The covering is characterized by its type ty, homology group hom (where 1 means Z), the number of cusps cp, the number of generators gens of the fundamental group, the Chern-Simons invariant CS and the type of link it represents (as identified in SnapPy). The links L7n1 (shown in Figure 3b) and L6a3 (shown in Figure 3c) correspond to the Hesse SIC and the two-qubit IC, respectively. The case of cyclic coverings corresponds to Brieskorn 3-manifolds as explained in the text: the spherical groups for these manifolds is given at the right hand side column.


For the irregular covering of degree 3 and first homology Z + Z, the fundamental group provided by SnapPy is *π*1(*M*3) = # *<sup>a</sup>*, *<sup>b</sup>*|*ab*−2*a*−1*b*<sup>2</sup> \$ that, of course, corresponds to a representative *H* of one of the two conjugacy classes of subgroups of index 3 of the modular group Γ, following the theory of [27]. The organization of cosets of *H* in the two-generator group *G* = # *<sup>a</sup>*, *<sup>b</sup>*|*a*2, *<sup>y</sup>*<sup>3</sup> \$ ∼= Γ thanks to the Coxeter-Todd algorithm (implemented in the software Magma [34]) results in the permutation group *P* = 3|(1, 2, 3),(2, 3), as in ([16] Section 3.1). This permutation group is also the one obtained from the congruence subgroup Γ0(2) ∼= *S*<sup>3</sup> of Γ (where *S*<sup>3</sup> is the three-letter symmetric group) whose fundamental domain is in ([16] Figure 1b). Then, the eigenstates of the permutation matrix in *S*<sup>3</sup> of type (0, 1, ±1) serve as magic/fiducial state for the Hesse SIC [15,16].

A similar reasoning applied to the irregular coverings of degree 4, and first homology Z + Z and <sup>Z</sup> <sup>2</sup> + Z leads to the recognition of congruence subgroups <sup>Γ</sup>0(3) and 4*A*0, respectively, behind the corresponding manifolds. It is known from ([16] Section 3.2) that they allow the construction of two-qubit minimal IC-POVMs. For degree 5, the equiangular 5-dit MIC corresponds to the irregular covering of homology <sup>Z</sup> <sup>3</sup> + Z and to the congruence subgroup 5*A*<sup>0</sup> in <sup>Γ</sup> (as in [16] Section 3.3).

Five coverings of degree 6 allow the construction of the (two-valued) 6-dit IC-POVM whose geometry contain the picture of Borromean rings ([16] Figure 2c). The corresponding congruence subgroups of Γ are identified in Table 1. The first, viz Γ(2), define a 3-manifold whose fundamental group is the same as the one of the link *L*8*n*3. The other three coverings leading to the 6-dit IC are the congruence subgroups *γ* , 3*C*0, Γ0(4) and Γ0(5).

#### **3. Quantum Information from Universal Knots and Links**

In the previous section, we found the opportunity to rewrite results about the existence and construction of *d*-dimensional MICs in terms of the three-manifolds corresponding to some degree *d* coverings of the trefoil knot *T*1. However, neither *T*<sup>1</sup> nor the manifolds corresponding to its covering are hyperbolic. In the present section, we proceed with hyperbolic (and universal) knots and links and display the three-manifolds behind the low dimensional MICs. The method is as above in Section 2 in the sense that the fundamental group of a 3-manifold *M*<sup>3</sup> attached to a degree *d*-covering is the one of a representative of the conjugacy class of subgroups of the corresponding index in the relevant knot or link.

## *3.1. Three-Manifolds Pertaining to the Figure-of-Eight Knot*

The fundamental group for the figure-of-eight knot *K*<sup>0</sup> is

$$
\pi\_1(S^3 \backslash K\_0) = \left\langle x, y \middle| y \* x \* y^{-1}xy = xyx^{-1}yx \right\rangle .
$$

and the number of *d*-fold coverings is in the list

$$\eta\_d(\mathbb{K}\_0) = \{1, 1, 1, 2, 4, \ 11, 9, 10, 11, 38, \dots\} \dots$$

Table 2 establishes the list of 3-manifolds corresponding to subgroups of index *d* ≤ 7 of the universal group *<sup>G</sup>* = *<sup>π</sup>*1(*S*<sup>3</sup> \ *<sup>K</sup>*0). The manifolds are labeled otet*Nn* in [25] because they are oriented and built from *N* = 2*d* tetrahedra, with *n* an index in the table. The identification of 3-manifolds of finite index subgroups of *G* was first obtained by comparing the cardinality list *ηd*(*H*) of the corresponding subgroup *H* to that of a fundamental group of a tetrahedral manifold in SnapPy table [28]. However, there is a more straightforward way to perform this task by identifying a subgroup *H* to a degree *d* covering of *K*<sup>0</sup> [27]. The full list of *d*-branched coverings over the figure eight knot up to degree 8 is available in SnapPy. Extra invariants of the corresponding *M*<sup>3</sup> may be found there. In addition, the lattice of branched coverings over *K*<sup>0</sup> was investigated in [35].

**Table 2.** Table of 3-manifolds *M*<sup>3</sup> found from subgroups of finite index *d* of the fundamental group *<sup>π</sup>*1(*S*<sup>3</sup> \ *<sup>K</sup>*0) (alias the *<sup>d</sup>*-fold coverings of *<sup>K</sup>*0). The terminology in column 3 is that of Snappy [28]. The identified *M*<sup>3</sup> is made of 2*d* tetrahedra and has cp cusps. When the rank *rk* of the POVM Gram matrix is *d*<sup>2</sup> the corresponding IC-POVM shows *pp* distinct values of pairwise products as shown.


Let us give more details about the results summarized in Table 2. Using Magma, the conjugacy class of subgroups of index 2 in the fundamental group *G* is represented by the subgroup on three generators and two relations as follows *H* = # *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*|*y*−1*zx*−1*zy*−1*x*<sup>−</sup>2, *<sup>z</sup>*−1*yxz*−1*yz*−1*xy*\$ , from which the sequence of subgroups of finite index can be found as *<sup>η</sup>d*(*M*3) = {1, 1, 5, 6, 8, 33, 21, 32, ···}. The manifold *M*<sup>3</sup> corresponding to this sequence is found in Snappy as otet0400002, alias *m*206.

The conjugacy class of subgroups of index 3 in *G* is represented as

$$H = \left\langle \mathbf{x}, y, z \middle| \mathbf{x}^{-2} z \mathbf{x}^{-1} y z^2 \mathbf{x}^{-1} z y^{-1}, z^{-1} \mathbf{x} z^{-2} \mathbf{x} z^{-2} y^{-1} \mathbf{x}^{-2} z y \right\rangle \mathbf{x}$$

with *<sup>η</sup>d*(*M*3) = {1, 7, 4, 47, 19, 66, 42, 484, ···} corresponding to the manifold otet0600003, alias *<sup>s</sup>*961.

As shown in Table 2, there are two conjugacy classes of subgroups of index 4 in *G* corresponding to tetrahedral manifolds otet0800002 (the permutation group *P* organizing the cosets is Z4) and otet0800007 (the permutation group organizing the cosets is the alternating group *A*4). The latter group/manifold has fundamental group

$$H = \left\langle \mathbf{x}, y, z | y \mathbf{x}^{-1} \mathbf{y}^{-1} \mathbf{z}^{-1} \mathbf{x} \mathbf{y}^{-2} \mathbf{x} \mathbf{y} \mathbf{z} \mathbf{x}^{-1} y, \mathbf{z} \mathbf{x}^{-1} y \mathbf{x}^{-1} y \mathbf{x}^{-1} z y \mathbf{x}^{-1} y^{-1} \mathbf{z}^{-1} \mathbf{x} \mathbf{y}^{-1} \right\rangle,$$

with cardinality sequences of subgroups as *<sup>η</sup>d*(*M*3) = {1, 3, 8, 25, 36, 229, 435 ···}. To *<sup>H</sup>* is associated an IC-POVM [15,16] which follows from the action of the two-qubit Pauli group on a magic/fiducial state of type (0, 1, −*ω*6, *ω*<sup>6</sup> − 1), with *ω*<sup>6</sup> = exp(2*iπ*/6) a six-root of unity.

For index 5, there are three types of 3-manifolds corresponding to the subgroups *H*. The tetrahedral manifold otet1000026 of cardinality sequence *<sup>η</sup>d*(*M*3) = {1, 7, 15, 88, 123, 802, 1328 ···}, is associated to a 5-dit equiangular IC-POVM, as in ([15] Table 5).

For index 6, the 11 coverings define six classes of 3-manifolds and two of them: otet1200041 and otet1200000 are related to the construction of ICs. For index 7, one finds three classes of 3-manifolds with two of them: otet1400002 (alias *L*14*n*55217) and otet1400035 are related to ICs. Finally, for index 7, 3 types of 3-manifolds exist, two of them relying on the construction of the 7-dit (two-valued) IC. For index 8, there exists 6 distinct 3-manifolds (not shown) none of them leading to an IC.

#### A Two-Qubit Tetrahedral Manifold

The tetrahedral three-manifold otet0800007 is remarkable in the sense that it corresponds to the subgroup of index 4 of *G* that allows the construction of the two-qubit IC-POVM. The corresponding hyperbolic polyhedron taken from SnapPy is shown in Figure 4a. Of the 29 orientable tetrahedral manifolds with at most 8 tetrahedra, 20 are two-colorable and each of those has at most 2 cusps. The 4 three-manifolds (with at most 8 tetrahedra) identified in Table 2 belong to the 20's and the two-qubit tetrahedral manifold otet0800007 is one with just one cusp ([37] Table 1).

**Figure 4.** Two platonic three-manifolds leading to the construction of the two-qubit MIC. Details are given in Tables 2 and 3.


**Table 3.** A few 3-manifolds *M*<sup>3</sup> found from subgroups of the fundamental group associated to the Whitehead link. For *<sup>d</sup>* <sup>≥</sup> 4, only the *<sup>M</sup>*3's leading to an IC are listed.

#### *3.2. Three-Manifolds Pertaining to the Whitehead Link*

One could also identify the 3-manifold substructure of another universal object, viz the Whitehead link *L*<sup>0</sup> [38].

The cardinality list corresponding to the Whitehead link group *π*1(*L*0) is

$$\eta\_d(L\_0) = \{1, 3, 6, 17, 22, 79, 94, 412, 616, 1659, \ldots\},$$

Table 3 shows that the identified 3-manifolds for index *d* subgroups of *π*1(*L*0) are aggregates of *d* octahedra. In particular, one finds that the qutrit Hesse SIC can be built from ooct0300014 and that the two-qubit IC-POVM may be built from ooct0400058. The hyperbolic polyhedron for the latter octahedral manifold taken from SnapPy is shown in Figure 4b. The former octahedral manifold follows from the link *L*12*n*1741 shown in Figure 5a and the corresponding polyhedron taken from SnapPy is shown in Figure 5b.

**Figure 5.** (**a**) The link *L*12*n*1741 associated to the qutrit Hesse SIC, (**b**) The octahedral manifold ooct0300014 associated to the 2-qubit IC.

#### *3.3. A Few Three-Manifolds Pertaining to Borromean Rings*

Three-manifolds corresponding to coverings of degree 2 and 3 of the 3-manifold branched along the Borromean rings *L*6*a*4 (that is a not a (3,3)-torus link but an hyperbolic link) (see Figure 1c) are given in Table 4. The identified manifolds are hyperbolic octahedral manifolds of volume 14.655 (for the degree 2) and 21.983 (for the degree 3).

**Table 4.** Coverings of degrees 2 to 4 branched over the Borromean rings. The identification of the corresponding hyperbolic 3-manifold *M*<sup>3</sup> is at the 5th column. Only two types of 3-manifolds allow the building of the Hesse SIC. The two 3-manifolds of degree 4 allow the construction of the two-qubit MIC to be identified by the cardinality structure of their subgroups/coverings.


## **4. A Few Dehn Fillings and Their POVMs**

To summarize our findings of the previous section, we started from a building block, a knot (viz the trefoil knot *T*1) or a link (viz the figure-of-eight knot *K*0) whose complement in *S*<sup>3</sup> is a 3-manifold *M*3. Then a *d*-fold covering of *M*<sup>3</sup> was used to build a *d*-dimensional POVM, possibly an IC. Now we apply a kind of 'phase surgery' on the knot or link that transforms *M*<sup>3</sup> and the related coverings while preserving some of the POVMs in a way to be determined. We will start with our friend *T*<sup>1</sup> and arrive at a few standard 3-manifolds of historic importance, the Poincaré homology sphere [alias the Brieskorn sphere Σ(2, 3, 5)], the Brieskorn sphere Σ(2, 3, 7) and a Seifert fibered toroidal manifold Σ . Then we introduce the 3-manifold Σ*<sup>Y</sup>* resulting from 0-surgery on the figure-of-eight knot [39]. Later in this section, we will show how to use the {3, 5, 3} Coxeter lattice and surgery to arrive at a hyperbolic 3-manifold Σ120*<sup>e</sup>* of maximal symmetry whose several coverings (and related POVMs) are close to the ones of the trefoil knot [40].

Let us start with a Lens space *L*(*p*, *q*) that is 3-manifold obtained by gluing the boundaries of two solid tori together, so that the meridian of the first solid torus goes to a (*p*, *q*)-curve on the second solid torus [where a (*p*, *q*)-curve wraps around the longitude *p* times and around the meridian *q* times]. Then we generalize this concept to a knot exterior, i.e., the complement of an open solid torus knotted like the knot. One glues a solid torus so that its meridian curve goes to a (*p*, *q*)-curve on the torus boundary of the knot exterior, an operation called Dehn surgery ([1] (p. 275), [24] (p. 259), [41]). According to Lickorish's theorem, every closed, orientable, connected 3-manifold is obtained by performing Dehn surgery on a link in the 3-sphere. For example, surgeries on the trefoil knot allow to build the most important spherical 3-manifolds—the ones with a finite fundamental group—that are the basis of ADE correspondence. The acronym ADE refers to simply laced Dynkin diagrams that connect apparently different objects such as Lie algebras, binary polyhedral groups, Arnold's theory of catastophes, Brieskorn spheres and quasicrystals, to mention a few [42].

### *4.1. A Few Surgeries on the Trefoil Knot*

#### The Poincaré Homology Sphere

The Poincaré dodecahedral space (alias the Poincaré homology sphere) was the first example of a 3-manifold not the 3-sphere. It can be obtained from (−1, 1) surgery on the left-handed trefoil knot *T*<sup>1</sup> [43].

Let *p*, *q*,*r* be three positive integers and mutually coprime, the Brieskorn sphere Σ(*p*, *q*,*r*) is the intersection in C<sup>3</sup> of the 5-sphere *S*<sup>5</sup> with the surface of equation *z p* <sup>1</sup> + *z q* <sup>2</sup> + *<sup>z</sup><sup>r</sup>* <sup>3</sup> = 1. The homology of a Brieskorn sphere is that of the sphere *S*3. A Brieskorn sphere is homeomorphic but not diffeomorphic to *S*3. The sphere Σ(2, 3, 5) may be identified to the Poincaré homology sphere. The sphere Σ(2, 3, 7) [39] may be obtained from (1, 1) surgery on *T*1. Table 5 provides the sequences *η<sup>d</sup>* for the corresponding surgeries (±1, 1) on *T*1. Plain digits in these sequences point out the possibility of building ICs of the corresponding degree. This corresponds to a considerable filtering of the ICs coming from *T*1.

**Table 5.** A few surgeries (column 1), their name (column 2) and the cardinality list of *d*-coverings (alias conjugacy classes of subgroups). Plain characters are used to point out the possible construction of an IC-POVM in at least one the corresponding three-manifolds (see [16] and Section 2 for the ICs corresponding to *T*1).


For instance, the smallest IC from Σ(2, 3, 5) has dimension five and is precisely the one coming from the congruence subgroup 5*A*<sup>0</sup> in Table 1. However, it is built from a non modular (fundamental) group whose permutation representation of the cosets is the alternating group *A*<sup>5</sup> ∼= (1, 2, 3, 4, 5),(2, 4, 3) (compare [15] Section 3.3).

The smallest dimensional IC derived from Σ(2, 3, 7) is 7-dimensional and two-valued, the same as the one arising from the congruence subgroup 7*A*<sup>0</sup> given in Table 1. However, it arises from a non modular (fundamental) group with the permutation representation of cosets as *PSL*(2, 7) ∼= (1, 2, 4, 6, 7, 5, 3),(2, 5, 3)(4, 6, 7).

## *4.2. The Seifert Fibered Toroidal Manifold* Σ

An hyperbolic knot (or link) in *S*<sup>3</sup> is one whose complement is 3-manifold *M*<sup>3</sup> endowed with a complete Riemannian metric of constant negative curvature, i.e., it has a hyperbolic geometry and finite volume. A Dehn surgery on a hyperbolic knot is exceptional if it is reducible, toroidal or Seifert fibered (comprising a closed 3-manifold together with a decomposition into a disjoint union of circles called fibers). All other surgeries are hyperbolic. These categories are exclusive for a hyperbolic knot. In contrast, a non-hyperbolic knot such as the trefoil knot admits a toroidal Seifert fiber surgery Σ obtained by (0, 1) Dehn filling on *T*<sup>1</sup> [44].

The smallest dimensional ICs built from Σ are the Hesse SIC that is obtained from the congruence subgroup Γ0(2) (as for the trefoil knot) and the two-qubit IC that comes from a non modular fundamental group [with cosets organized as the alternating group *A*<sup>4</sup> ∼= (2, 4, 3),(1, 2, 3)].

## *4.3. Akbulut's Manifold* Σ*<sup>Y</sup>*

Exceptional Dehn surgery at slope (0, 1) on the figure-of-eight knot *K*<sup>0</sup> leads to a remarkable manifold Σ*<sup>Y</sup>* found in [39] in the context of 3-dimensional integral homology spheres smoothly bounding integral homology balls. Apart from its topological importance, we find that some of its coverings are associated to already discovered ICs and those coverings have the same fundamental group *π*1(Σ*Y*).

The smallest IC-related covering (of degree 4) occurs with integral homology Z and the congruence subgroup Γ0(3) also found from the trefoil knot (see Table 1). Next, the covering of degree 6 and homology <sup>Z</sup> <sup>5</sup> + Z leads to the 6-dit IC of type 3*C*<sup>0</sup> (also found from the trefoil knot). The next case corresponds to the (non-modular) 11-dimensional (3-valued) IC.

## *4.4. The Hyperbolic Manifold* Σ120*<sup>e</sup>*

The hyperbolic manifold closest to the trefoil knot manifold known to us was found in [40]. The goal in [40] is the search of—maximally symmetric—fundamental groups of 3-manifolds. In two dimensions, maximal symmetry groups are called Hurwitz groups and arise as quotients of the (2, 3, 7)-triangle groups. In three dimensions, the quotients of the minimal co-volume lattice Γ*min* of hyperbolic isometries, and of its orientation preserving subgroup Γ<sup>+</sup> *min*, play the role of Hurwitz groups. Let *C* be the {3, 5, 3} Coxeter group, Γ*min* the split extension *C* - Z<sup>2</sup> and Γ<sup>+</sup> *min* one of the index two subgroups of Γ*min* of presentation

$$
\Gamma\_{\rm min}^+ = \left\langle \mathbf{x}, \mathbf{y}, \mathbf{z} | \mathbf{x}^3, \mathbf{y}^2, \mathbf{z}^2, (\mathbf{x}yz)^2, (\mathbf{x}zyz)^2, (\mathbf{x}y)^5 \right\rangle \dots
$$

According to ([40] Corollary 5), all torsion-free subgroups of finite index in Γ<sup>+</sup> *min* have index divisible by 60. There are two of them of index 60, called Σ60*<sup>a</sup>* and Σ60*b*, obtained as fundamental groups of surgeries *<sup>m</sup>*017(−4, 3) and *<sup>m</sup>*016(−4, 3). Torsion-free subgroups of index 120 in <sup>Γ</sup><sup>+</sup> *min* are given in Table 6. It is remarkable that these groups are fundamental groups of oriented three-manifolds built with a single icosahedron except for Σ120*<sup>e</sup>* and Σ120*g*.

**Table 6.** The index 120 torsion-free subgroups of Γ<sup>+</sup> *min* and their relation to the single isosahedron 3-manifolds [40]. The icosahedral symmetry is broken for Σ120*<sup>e</sup>* (see the text for details).


The group Σ120*<sup>e</sup>* is special in the sense that many small dimensional ICs may be built from it in contrast to the other groups in Table 6. The smallest ICs that may be built from Σ120*<sup>e</sup>* are the Hesse SIC coming from the congruence subgroup Γ0(2), the two-qubit IC coming the congruence subgroup 4*A*<sup>0</sup> and the 6-dit ICs coming from the congruence subgroups Γ(2), 3*C*<sup>0</sup> or Γ0(4) (see [16] Section 3 and Table 1). Higher dimensional ICs found from Σ120*<sup>e</sup>* do not come from congruence subgroups.

#### **5. Conclusions**

The relationship between 3-manifolds and universality in quantum computing has been explored in this work. Earlier work of the first author already pointed out the importance of hyperbolic geometry and the modular group Γ for deriving the basic small dimensional IC-POVMs. In Section 2, the move from Γ to the trefoil knot *T*<sup>1</sup> (and the braid group *B*3) to non-hyperbolic 3-manifolds could be

investigated by making use of the *d*-fold coverings of *T*<sup>1</sup> that correspond to *d*-dimensional POVMs (some of them being IC). Then, in Section 3, we went on to universal links (such as the figure-of-eight knot, Whitehead link and Borromean rings) and the related hyperbolic platonic manifolds as new models for quantum computing based POVMs. Finally, in Section 4, Dehn fillings on *T*<sup>1</sup> were used to explore the connection of quantum computing to important exotic 3-manifolds (i.e., Σ(2, 3, 5) and Σ(2, 3, 7)), to the toroidal Seifert fibered Σ , to Akbulut's manifold Σ*<sup>Y</sup>* and to a maximum symmetry hyperbolic manifold Σ120*<sup>e</sup>* slightly breaking the icosahedral symmetry. It is expected that our work will have importance for new ways of implementing quantum computing and for the understanding of the link between quantum information and cosmology [45–47]. A subsequent paper of ours develops the field of 3-manifold based UQC with its relationship to Bianchi groups [48].

**Author Contributions:** All authors contributed significantly to the content of the paper. M.P. wrote the manuscript and the co-authors reviewed it.

**Funding:** The first author acknowledges the support by the French "Investissements d'Avenir" program, project ISITE-BFC (contract ANR-15-IDEX-03). The other resources came from Quantum Gravity Research.

**Conflicts of Interest:** The authors declare no competing interests.

## **References**


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