Phase operator on L^2(\mathbb{Q}_p) and the zeroes of Fisher and Riemann
aa r X i v : . [ m a t h - ph ] F e b Phase operator on L ( Q p ) andthe zeroes of Fisher and Riemann Parikshit Dutta ∗ and Debashis Ghoshal † Asutosh College, 92 Shyama Prasad Mukherjee Road,Kolkata 700026, India School of Physical Sciences, Jawaharlal Nehru University,New Delhi 110067, India
Abstract
The distribution of the non-trivial zeroes of the Riemann zeta function, accordingto the Riemann hypothesis, is tantalisingly similar to the zeroes of the partitionfunctions (Fisher and Yang-Lee zeroes) of statistical mechanical models studied byphysicists. The resolvent function of an operator akin to the phase operator, conju-gate to the number operator in quantum mechanics, turns out to be important inthis approach. The generalised Vladimirov derivative acting on the space L ( Q p ) ofcomplex valued locally constant functions on the p -adic field is rather similar to thenumber operator. We show that a ‘phase operator’ conjugate to it can be constructedon a subspace L ( p − Z p ) of L ( Q p ) . We discuss (at physicists’ level of rigour) howto combine this for all primes to possibly relate to the zeroes of the Riemann zetafunction. Finally, we extend these results to the family of Dirichlet L -functions, us-ing our recent construction of Vladimirov derivative like pseudodifferential operatorsassociated with the Dirichlet characters. ∗ [email protected] † [email protected] Introduction
The statistical distribution of the zeroes of the Riemann zeta function, and the relatedfamily of Dirichlet L -functions, qualitatively resemble the eigenvalue distribution of arandom ensemble of unitary matrices [1–3]. It is also reminiscent of the distribution ofzeroes of partition functions of statistical models. The latter observation is the motivationto search for a suitable model in physicists’ approach to the problem—the literature isvast, however, see e.g., Refs. [4–9], the review [10] and references therein. This resemblancemay be an important guide as the zeroes of the partition function of many systems, by theYang-Lee type theorems [11], all lie parallel to the imaginary axis (or on the unit circle).These zeroes are called Yang-Lee zeroes or Fisher zeroes, depending upon whether thepartition function is viewed as a function of the applied external field, e.g., magnetic field,or of β = 1 / ( k B T ) , the inverse temperature. The arithmetic or primon gas of Refs. [5–7]and the number theoretic spin chain of Refs. [8, 9], in particular, proposed interestingmodels for which the partition functions are directly related to the Riemann zeta function.In this approach the non-trivial zeroes of the zeta function are to be identified with theYang-Lee or Fisher zeroes.Motivated by these, we shall propose a statistical model and compute its partitionfunction. The idea is again to associate the spectrum of an operator with the Fisherzeroes of the partition function. In addition, however, we shall study the spectrum ofsome relevant operators of these models. The systems that relate to the L -functions ofour interest can be thought of as spins in an external magnetic field. Since the spectrum ofa Hamiltonian of this type of spin systems is discrete (the spins being integer/half-integervalued) this operator is similar to the number operator of an oscillator. A phase operatorthat is conjugate to this will be a new ingredient in our investigation. The construction of aphase operator which is truly canonically conjugate to the number operator is a subject oflong-standing quest that may not be completely closed yet. Nevertheless, several differentways to define the phase operator have been proposed, for example, Refs. [12–18] is apartial list. In particular, we shall investigate two ways of defining it for the spin modelscorresponding to the family of L -functions. In the first construction, we follow Ref. [17],where the authors propose an operator by directly constructing eigenstates of phase fora system with a discrete spectrum. The second approach is motivated by the proposalin [15]. We shall argue that there are enough hints in these proposals to understand thecorrespondence between the spectrum of these operators and the zeroes of the partitionfunction.In the following, we shall first review (in Section 2) some of the relevant argumentsand results from the cited references, in the context of a simple spin system on a one-dimensional lattice. In Section 3, after recalling some properties of the Riemann zetafunction and our earlier work on its relation to operators on the Hilbert space of complexvalued functions on the p -adic number field Q p [19, 20], we elaborate on a proposal toview it as a statistical model of spins. In Sections 3.1 and 3.2 we detail two constructionsof the phase operators for the spin model for the Riemann zeta function, which are thenextended to the family of Dirichlet L -functions in Section 4.2 Quantum spins in external field
Spin models in one dimension are among the simplest statistical models, yet they offeran arena rich enough to experiment, before considering more complicated systems. Thevariables are ‘spins’ s n at lattice points n ∈ Z or n ∈ N that can take (2 j + 1) values {− j, − j + 1 , · · · , j − , j } in the spin- j representation. In models of magnetism, thesespins interact locally, usually with the nearest neighbours. In addition, one may turn onan external magnetic field.Let us digress to recall the properties of a simpler model, the Ising model, in whichthe classical spins take one of two possible values ± and the total Hamiltonian is H = − J P n s n s n +1 − B P n s n , where J is the strength of interaction ( J > being ferromagneticand anti-ferromagnetic otherwise) and the second term arises from an interaction with anexternal magnetic field B . The partition function (in the absence of an external field) ofan Ising system of size L at inverse temperature β is Z ( β ) ≡ Tr e − βH = X { s n } exp βJ L − X n =1 s n s n +1 ! In this simple case, one may also change variables to σ n ≡ σ h n − ,n i = s n − s n associated tothe edges h n − , n i joining nearest neighbours. Evidently, σ n = ± as well. Thus Z ( β ) = 2 X { σ n } exp βJ L X n =2 σ n ! = 2 X σ n h σ , σ , · · · | exp βJ X j S j ! | σ , σ , · · · i where we have defined vectors | σ n i in a (two-dimensional) Hilbert space corresponding tothe edge h n − , n i and S n s are spin operators such that S n | σ n i = σ n | σ n i . A generalisationof this model allows the coupling constants J to be position dependent, so that theHamiltonian is H = − P n J n σ n and Z ( β ) = 2 X σ n (cid:10) σ , σ , · · · (cid:12)(cid:12) e β P n J n S n (cid:12)(cid:12) σ , σ , · · · (cid:11) is the canonical partition function of the generalised model at the temperature k B T = 1 /β .We would like to consider the general case where the spins to be valued in the spin- j representation of su (2) . Although we seek a partition function of the form as above, thegeneral spin case is cannot be realised as an Ising type model, rather it will be a modelof spins in an external local magnetic field B n at site n . It will be useful to think of S n to be the third component S n of the su (2) spin operators on the edge/site, the othersbeing S n ± . The vectors | σ , σ , · · · i = | σ i ⊗ | σ i ⊗ · · · belong to the product space.Theinteraction ∼ B · S between the spin (magnetic moment to be precise) and the externalfield (assumed to be along the z -direction) described by the Hamiltonian H = − P n B n σ n Z ( β ) = X σ n (cid:10) σ , σ , · · · (cid:12)(cid:12) e β P n B n S n (cid:12)(cid:12) σ , σ , · · · (cid:11) at the temperature k B T = 1 /β . Our objective is to obtain an identity for the partitionfunction for this model. To this end, we shall seek an operator that, in a certain well de-fined sense, is formally canonically conjugate to the z -component S n of the spin operatorat site n . There are well known difficulties in defining such an operator, however, we shallsee that one needs to make a much weaker demand.In this context, it is useful to remeber the Schwinger oscillator realisation of the algebra su (2) in terms of a pair of bosonic creation/anhilation operators ( a † , a , a † , a ) at eachedge, where we drop the edge index for the time being. Then S + = a † a , S − = a † a and the third component is the difference of the number operators S = ( n − n ) = (cid:16) a † a − a † a (cid:17) . One can formally introduce the phase operator Φ = ( φ − φ ) suchthat [ φ a , n b ] = iδ ab , however, there are several mathematical difficulties in defining theabove [12,13]. We will now review an explicit construction to show how one can still workaround this problem. Let us label the eigenstates in the spin- j representation of S as | m i , for m = − j, · · · , j .One can define an eigenstate of phase as a unitary transform of these states as | φ k i = 1 √ j + 1 j X m = − j e − imφ k B | m i where, φ k = 2 πkB (2 j + 1) , k = − j, · · · , j (1)are the eigenvalues of the phase. The phase eigenstates satisfy h φ k ′ | φ k i = 12 j + 1 j X m = − j e − imB ( φ k − φ k ′ ) = δ k,k ′ (2)and thus provide an orthonormal basis of the Hilbert space of states.In terms of these, we may define the the ‘phase operator’ through spectral decompo-sition as ˆ φ = j X k = − j φ k | φ k ih φ k | (3)We shall now show that it transforms covariantly when conjugated by e βBS . This worksfor special values of β since, an eigenvalues φ k of ˆ φ , being angle-valued, is only defined4odulo π/B . In order to see this, we note that e − βBS ˆ φ e βBS = j X k = − j φ k j + 1 j X m = − j j X m ′ = − j e − im ( φ k − iβ ) B + im ′ ( φ k − iβ ) B | m ih m ′ | There are two cases to consider. The first is trivial: for β = 0 or any of its periodic images iβ = πnB ( n ∈ Z ) in the complex β -plane, the RHS is the phase operator ˆ φ . More inter-estingly, if iβ takes any of the specific discrete values πk ′ B (2 j +1) + πnB , where k ′ = − j, · · · , j (but k ′ = 0 ) and n ∈ Z , i.e., iβ is a difference between the phase eigenvalues (mod π/B ),then φ k − iβ is again an allowed eigenvalue of the phase operator (mod 2 π/B ) . In thiscase, we can add and subtract iβ to the eigenvalue φ k and use the completeness of basis,to find e − βBS ˆ φ e βBS = ˆ φ + iβ only for = β = − πijB (2 j + 1) , · · · , πijB (2 j + 1) (cid:16) mod πB (cid:17) (4)This is called a shift covariance relation [21]. It may also be rewritten as a commutator (cid:2) ˆ φ, e βBS (cid:3) = iβ e βBS only for = β = − πijB (2 j + 1) , · · · , πijB (2 j + 1) (cid:16) mod πB (cid:17) i.e., at special values of the inverse temperature.To summarise, we find that ˆ φ in Eq. (3) satisfies shift covariance, alternatively, thoughsomewhat loosely, it is ‘canonically conjugate’ to S only for a special set of an infinitenumber of imaginary values of β , all on the line Re β = 0 as above. At β = 0 (mod πB ),the commutator is trivial .In passing, it is instructive to take the trace of the ‘canonical commutator’. The lefthand side evidently vanishes, since the vector space of states is finite, namely (2 j + 1) ,dimensional. On the right hand side, the trace of e − βH , the partition function whichvanishes, being a sum over the roots of unity. Thus the values of β for which Eq. (4)is valid must also satisfy the condition Tr e − βH = 0 . This means that mod π/B , thesevalues of iβ = 0 for which the partition function has a zero are same as that of theeigenvalues of ˆ φ .The resolvent of the exponential of the phase operator at a single site (as a functionof z = e iφ ) is ˆ R [ ˆ φ ]( φ ) = (cid:16) − e − iφ e i ˆ φ (cid:17) − and its trace is Tr (cid:16) ˆ R [ ˆ φ ]( φ ) (cid:17) = j X k = − j D φ k (cid:12)(cid:12)(cid:12) − e i ( ˆ φ − φ ) (cid:12)(cid:12)(cid:12) φ k E = j X k = − j − e i ( φ k − φ ) On the other hand, the partition function at a single site Z ( β ) = Tr e − βH = P m e βBm vanishes at special values of the inverse temperature β = πmiB (2 j +1) (mod πB ) where m ∈ It is also reflected in the resolvent of the phase operator, as we shall see in the following. − j, · · · , j } but m = 0 . These zeroes of the partition function in the complex β -planeare called Fisher zeroes . At precisely these values, the resolvent function develops poles.
Before we get to our main goal to interpret the Riemann zeta function as a partitionfunction, let us briefly recall some of its relevant properties. Originally defined by theanalytical continuation of the series ζ ( s ) = ∞ X n =1 n s = Y p ∈ primes − p − s ) , Re( s ) > (5)to the complex s -plane by Riemann, the zeta function has a set of equally spaced zeroesat negative even integers − n , n ∈ Z called its trivial zeroes. More interestingly, ithas another infinite set of zeroes, which, according to the Riemann hypothesis lie on the critical line
Re( s ) = . The related Riemann ξ -function (sometimes called the symmetriczeta-function ) and the adelic zeta function share only the latter (non-trivial) zeroes withEq. (5) (i.e., the set of trivial zeroes are absent in the following functions) ξ ( s ) = 12 s ( s − ζ A ( s ) = 12 s ( s − π − s Γ (cid:16) s (cid:17) ζ ( s ) (6)both of which satisfy the reflection identity ξ ( s ) = ξ (1 − s ) , respectively, ζ A ( s ) = ζ A (1 − s ) ,derived from a similar identity for the original zeta function. The former is a holomorphicfunction while the latter, ζ A ( s ) , is meromorphic.The non-trivial zeroes of ζ ( s ) (which are the only zeroes of ξ ( s ) and ζ A ( s ) ) conjecturallyon the critical line, seem to occur randomly, although they are found to be correlated inthe same way as the eigenvalues of a Gaussian ensemble of N × N hermitian or unitarymatrices in the limit N → ∞ [1–3]. Starting from Hilbert and Pólya, it has long beenthought that these zeroes correspond to the eigenvalues of an operator, that is self-adjointin an appropriately defined sense. A direct analysis of the spectrum of the purportedoperator may lead to a proof of the Riemann hypothesis. Despite many ingenious efforts,an operator has not yet been found. In Ref. [19], in a larger collaboration, we attemptedto find a suitable operator by assuming the validity of the hypothesis , specifically, byassuming that the zeroes are the eigenvalues of a unitary matrix model (UMM). Wefound that the partition function can be expressed as the trace of an operator on theHilbert space of complex valued locally constant Bruhat-Schwarz functions supported ona compact subset p − Z p of the p -adic field Q p . This was achieved in two steps. Firsta UMM was constructed for each prime p corresponding to the Euler product form inEq. (5). These (as well as a UMM for the trivial zeroes) were combined to define therandom matrix model. In this paper, we shall use some of the technology that were usefulin [19], however, our goal will be different. 6e begin by expanding the prime factors in the Euler product form of the zeta function ζ ( s ) = Y p ∈ primes − p − s ) = Y p ∈ primes ∞ X n ( p ) =0 p − sn ( p ) , Re( s ) > (7)The factor ζ p ( s ) = 1(1 − p − s ) for a fixed prime p (8)is sometimes called the local zeta function at p . It can be thought of as a complex valuedfunction on the field Q p (of p -adic numbers). The prefactor ζ R ( s ) = π − s Γ (cid:0) s (cid:1) in Eq. (6)is known as the local zeta functions corresponding to R (of real numbers). It is theMellin transform of the Gaussian function e − πx . In an exactly analogous fashion, ζ p ( s ) in Eq. (8) is the Mellin transform of the equivalent of the Gaussian function (in the senseof a function that is its own Fourier transform) on Q p .We can express the sum in Eq. (7) as the trace of an operator. To this end, let usrecall that the space of (mean-zero) square integrable complex valued functions on Q p isspanned by the orthonormal set of Kozyrev wavelets ψ ( p ) nml ( ξ ) ∈ C (for ξ ∈ Q p ), whichhave compact support in Q p [22]. In p segments (of equal Haar measure) its values arethe p -th roots of unity. They are analogous to the generalised Haar wavelets, with thelabels n , m and l referring to scaling, translation and phase rotation. Interestingly, theKozyrev wavelets are eigenfunctions of an operator with eigenvalue p α (1 − n ) D α ( p ) ψ ( p ) n,m,l ( ξ ) = p α (1 − n ) ψ ( p ) n,m,l ( ξ ) (9)where, the pseudodifferential operators D α ( p ) , called the generalised Vladimirov derivatives ,are defined by the following integral kernel as D α ( p ) f ( ξ ) = 1 − p α − p − α − Z Q p dξ ′ f ( ξ ′ ) − f ( ξ ) | ξ ′ − ξ | α +1 p , α ∈ C They satisfy D α ( p ) D α ( p ) = D α ( p ) D α ( p ) = D α + α ( p ) .Since the roles of translation and phase are not going to be important in what follows,let us set m = 0 and l = 1 and define vectors | n ( p ) i corresponding to ψ ( p ) − n +1 , , ( ξ ) ψ ( p ) − n +1 , , ( ξ ) ←→ | n ( p ) i (10)in the Hilbert space L ( Q p ) . Then D α ( p ) | n ( p ) i = p n ( p ) α | n ( p ) i log p D ( p ) | n ( p ) i = lim α → D α ( p ) − α ln p | n ( p ) i = n ( p ) | n ( p ) i (11)The wavelets, by construction, transform naturally under the affine group of scaling andtranslation. However, it was shown in Ref. [23] that the scaling part of it enhances toa larger SL(2, R ) symmetry. In terms of the raising and lowering operators a ( p ) ± | n ( p ) i = n ( p ) ± i the generators of SL(2, R ) are J ( p ) ± = a ( p ) ± log p D ( p ) and J ( p )3 = log p D ( p ) . Thealgebra of these generators and their action on the wavelet states are as follows. h J ( p )3 , J ( p ) ± i = ± J ( p ) ± , h J ( p )+ , J ( p ) − i = − J ( p )3 J ( p )3 | n ( p ) i = n ( p ) | n ( p ) i , J ( p ) ± | n ( p ) i = n ( p ) | n ( p ) ± i (12)We can now write Eq. (7) as ζ ( s ) = Y p ∈ primes ∞ X n ( p ) =0 (cid:10) n ( p ) (cid:12)(cid:12) D − s ( p ) (cid:12)(cid:12) n ( p ) (cid:11) = X n =( n (2) ,n (3) , ··· ) h n | e − s ln D | n i (13)where we have used a shorthand ln D ≡ P p ln p log p D ( p ) , and the vectors | n i belong tothe product of the Hilbert spaces for all primes N p L ( Q p ) . However, since the sum runsonly over the positive integers (including zero), this subspace is actually N p L ( p − Z p ) ,spanned by the Bruhat-Schwarz functions restricted to p − Z p due to which the trace is welldefined (see [22–24] for details on the wavelet functions). This expression leads us to thinkof the zeta function as the partition function of a statistical system, the configurations ofwhich are parametrised by the integers n = ( n (2) , n (3) , · · · ) .The sl ( R ) algebra Eq. (12) can be realised in terms of a pair of oscillators in theSchwinger representation J ( p )3 = log p D ( p ) = 12 (cid:0) N I( p ) − N II(p) (cid:1) , J ( p )+ = a † I( p ) a II( p ) and J ( p ) − = a † II( p ) a I( p ) (14)Formally there is a phase difference operator Φ ( p ) = (cid:16) Φ ( p )I − Φ ( p )II (cid:17) conjugate to the numberdifference operator N ( p ) = (cid:0) N I( p ) − N II( p ) (cid:1) , such that [Φ I ( p ) , N J ( p ′ ) ] = iδ IJ δ pp ′ , [Φ ( p ) , N ( p ′ ) ] = iδ pp ′ (15)In Section 2 we reviewed a construction for the phase operator following [12–18]. Assumingfor the moment that a phase operator with the desired properties can be constructed, wedefine the operator p Φ ( p ) p − N ( p ) = p Φ ( p ) e − N ln p and evaluate the following commutator (cid:20) p Φ p p − N ( p ) , p ′ N ( p ′ ) (cid:21) = iδ pp ′ (16)using Eq. (15). Thus, the operator p Φ ( p ) p − N ( p ) is formally canonically conjugate to p N ( p ) = D ( p ) .We would now like to extend it to the large Hilbert space obtained by combining allprimes. Let us first consider all prime numbers up to a fixed prime p . The number of8uch primes is π ( p ) , where π ( x ) is the prime counting function. We now define O p = 1 π ( p ) p X p =2 p Φ ( p ) D − p and ln D p = p X p =2 ln D ( p ) which are operators in the truncated Hilbert space p O p =2 L ( p − Z p ) . These are canonicallyconjugate since [ O p , D p ] = i . Now we take the limit p → ∞ to obtain the canonicallyconjugate operators O = lim p →∞ O p , D = lim p →∞ D p such that [ O , D ] = i on the large Hilbert space N p L ( p − Z p ) . This limit is analogous to the thermodynamiclimit of statistical models, as we shall see in Section 3.1.Associated to these operators is the Weyl symmetric product
12 ( DO + OD ) = lim p →∞
12 ( D p O p + O p D p ) = OD − i
2= lim p →∞ π ( p ) p X p =2 (cid:18) ⊗ · · · ⊗ e ln D ( p ) Φ ( p ) ln p e − ln D ( p ) ⊗ ⊗ · · · (cid:19) (17)which is (formally) self-adjoint. In the last line, we have a similarity transform of the sumof the Φ ( p ) operators. As has been emphasised, e.g. in Ref. [15], the operator canonicallyconjugate to the number operator can only be defined up to a similarity transformation.Hence there ought to be more than one (which could be infinite in number) total phaseoperators Φ canonically conjugate to the total number operator ln D = P p ln D ( p ) . Onemay follow proposals in the literature (e.g. [15]) to define Φ ( p ) , which would result in ( D p O p + O p D p ) , canonically conjugate to ln D p , on the outer product of a dense sub-space of the Hilbert space L ( p − Z p ) at the p -th place. It is worth reiterating that theconstruction discussed above is formal. The limit p → ∞ is not straightforward. Thereis a more convenient way to construct the phase operator over a subspace of the Hilbertspace. We shall attempt to do so in the next two subsections. Let us return to the model of su (2) spin in an external field of Section 2 with the Hamil-tonian containing a site dependent magnetic field H = − p X p =2 B p N p = − p X p =2 B p ( S ,p + j ) of using prime numbers p to labelthe sites, B p are the values of the magnetic field at site p and we have shifted the zeroof the energy for convenience. The latter amounts to a shift in the spectrum of S p, by S p, → N p = S ,p + j so that N p takes the integer value , , · · · , n .In this case one can define a phase operator at an individual site, say φ p at the p -th site, as in Section 2. Each of these individual operators satisfies the shift covariancerelation (or commutator) for special values of β (cid:2) φ p , e β P p B p N p (cid:3) = iβe β P p B p N p (18)for β = 2 πikB p ( n + 1) (mod 2 π/B p ) with k = 1 , · · · , n and p = 2 , · · · , p This is valid over the entire Hilbert space, i.e., on an arbitrary state vector, but only forthese special values of β . Thus there are as many shift covariant phase operators as thenumber of sites, and each individual phase operator is covariant under the specific choicesof β . Moreover, since each of the Hilbert spaces, labelled by p , is finite dimensional, thetrace is a product over traces in each Hilbert space. Hence if we take the trace of Eq. (18),exactly as in the case of the spin model in Section 2, the trace of the commutator is zero,therefore, Tr e β P p B p N p = 0 . Thus the shift covariance relation is valid for those valuesof β which also satisfy the zero trace condition. This relates the zeroes of the partitionfunction to the poles of the following resolvent operators R [ e i φ p ]( φ ) = (cid:0) − e − iφ e i φ p (cid:1) − for all p . The trace of the resolvent is n X k p =0 − e − iφ + iφ kp apart from the pole for k = 0 , which yields the trivial commutator.The similarity between the spin in a magnetic field and the zeta function is apparentat this stage. (Recall that we have labelled the edges connecting adjacent sites by thefirst p prime numbers with this objective.) Indeed, if we choose the local magnetic field B p = ln p , then the partition function becomes Z ( β ) = p Y p =2 (cid:18) n X m p =0 e βm p ln p (cid:19) = p Y p =2 − p β ( n +1) − p β In the thermodynamic limit p → ∞ , even for finite n , the partition function has a simpleform in terms of a ratio of the Riemann zeta functions Z ( β ) = lim p →∞ p Y p =2 − p β ( n +1) − p β = ζ ( − β ) ζ ( − ( n + 1) β ) (19) It should, however, be mentioned that this type of numbering has been used before in [4–7]. κ -parafermionicprimon gas of [5–7, 10] with κ = n + 1 and s = − β . It would be interesting to try torelate the parafermionic variables to the spin degrees of freedom. Notice that Z ( β ) haszeroes at the non-trivial zeroes of ζ ( − β ) from the numerator, as well as at β = − / ( n + 1) from the pole of ζ ( − ( n + 1) β ) from the denominator. The latter is the only real zero,although it is at an unphysical value of the (inverse) temperature. However, the trivialzeroes of ζ ( − β ) are not zeroes of the partition function. This is due to the fact that atthese points, both the numerator and the denominator have simple zeroes, hence lim β → n ζ ( − β ) ζ ( − ( n + 1) β ) = finiteThus the nontrivial zeroes are the Fisher zeroes of the spin model in the complex (inversetemperature) β -plane. However, since the zeroes of the Riemann zeta function are believedto be isolated (and since there is no accumulation point on the real line) these zeroes arenot related to any phase transition. This is consistent as the system of spins in a magneticfield is not expected to undergo a phase transition. Finally, the partition function hasadditional poles from the zeroes of the zeta function in the denominator.The spectrum of the zeroes of the partition function is then given by p X p =2 X n p ∈ Z n X k p =0 − e iφ kp + i πnp ln p − iφ − X p,n p − e i πnp ln p − iφ (20)where we have subtracted the pole due to k = 0 . This function may be rewritten asfollows. p X p =2 X n ∈ Z − e i ( πn ( n +1) ln p − φ ) − p X p =2 X n ∈ Z − e i ( πn ln p − φ ) sing. ≈ p X p =2 X n ∈ Z − iφ − πn ( n +1) ln p − p X p =2 X n ∈ Z − iφ − πn ln p ≈ p X p =2 dd ( iφ ) ln(1 − p − ( n +1) iφ ) − dd ( iφ ) p X p =2 ln(1 − p − iφ ) ≈ − i ddφ ln (cid:18) p Y p =2 − p − ( n +1) iφ − p − iφ (cid:19) In the above we have used the Mittag-Leffler expansion, assuming analyticity of thepartition function. The expression above, in the limit p → ∞ , becomes − i ddφ ln (cid:18) ζ ( iφ ) ζ (( n + 1) iφ ) (cid:19) for Re ( iφ ) > .We will now try to construct a single operator that can be understood as ‘canonically11onjugate’ to the Hamiltonian. If we define the total phase operator as Φ = P p φ p (which is the sum of individual phase operators φ p as defined in Section 2) it does not,unfortunately, have the desired shift covariance relation Eq. (4) with the Hamiltonian.This is due to the site dependence of the magnetic field B p , as is apparent from the stepsleading to Eq. (4). The commutator there is obtained only at specific discrete values of β which are integer multiples of πk ′ /B p ( n + 1) . Therefore, unless the magnetic field B p at all the sites are commensurate, which is certainly not the case for B p = ln p , it is notpossible to get the desired commutator this way.Instead, we propose to work with an aggregate phase operator ϕ such that its actionon the composite state N p (cid:12)(cid:12) φ p,k p (cid:11) is defined to be e i ˆ φ p if the eigenvalue φ p = 0 , while allother eigenvalues φ q = p are zero, otherwise it acts as the identity. Thus, if two or more ofthe phases are non-zero , e i ϕ = . This may be expressed as e i ϕ = p X p =2 e i ˆ φ p Y q = p δ φ q , + 2 n =0 ( n =0 − p X p ,p =1 Y p = p (1 − δ φ p , )(1 − δ φ p , ) (21)where n =0 = P p (1 − δ φ p , ) is the number of sites where the phase is non-zero. This isequivalent to projecting on a subspace H (1) of the Hilbert space, in which only one, andexactly one, phase is different from zero . After the projection, one can use the total phaseoperator in the subspace Φ | H (1) = Π H (1) (cid:16)P p Φ p (cid:17) Π H (1) . From either point of view, theaction of the above is nontrivial on a subspace of the Hilbert space parametrised by onlyone of the eigenvalues φ p at a time, i.e., on a union of circles ∪ p S p ) while the full Hilbertspace is parametrised by ( S ) p . In the complement of this subspace, it is identity. In thissubspace H (1) , we can follow the steps leading to Eq. (4) to compute the commutator (cid:2) ϕ , Π H (1) e − βH Π H (1) (cid:3) = iβ Π H (1) e − βH Π H (1) which holds in H (1) for all β = πkB p ( n +1) (mod π/B p ) where k = 1 , · · · , n and p = 2 , · · · , p .It is worth emphasising that, as in several examples in quantum theory, the domain ofthe canonical commutator is not the entire Hilbert space, but a direct sum of closedorthogonal subspaces [14] of the type H (1) . In the limit p → ∞ , one take the closure ofthis subspace to obtain a closed subspace of the entire Hilbert space.The resolvent function of the exponential of the aggregate phase operator Eq. (21) R [ e i ϕ ]( φ ) = (cid:0) − e − iφ e i ϕ (cid:1) − (22)has the trace Tr (cid:0) R [ e i ϕ ]( φ ) (cid:1) = p X p =2 X k , ··· ,k p (cid:0) ⊗ p i =1 (cid:10) φ k i (cid:12)(cid:12)(cid:1) e − βH ∞ X n =0 e in ( ϕ − φ ) (cid:0) ⊗ p i =1 (cid:12)(cid:12) φ k i (cid:11)(cid:1) This is analogous, though not exactly equivalent, to a projection of the Fock space of a quantum fieldtheory of, say a scalar field, on a subspace with single-particle excitation. p X p =2 X kp exactly one φkp =0 X n =0 e inφ kp e − inφ + X kp at least two φkp =0 e − inφ ! = p X p =2 X kp exactly one φkp =0 − e iφ kp − iφ + X kp at least two φkp =0 − e − iφ ! (23)Except for the pole at φ = 0 , this behaviour is in fact identical to that of the resolvent p X p =2 (cid:16) − e − iφ e i ˆ φ p (cid:17) − in Eq. (20).Even though we do not require to take the limit n → ∞ , it is interesting to note thatthe phase operator φ p at the p -th site approaches the phase operator described in [15],which is a Toeplitz operator [25–27] in this limit. This has been shown in [17]. Thisprovides a way to understand the relation to the spectrum without a truncation to afinite n case. As shown in [15, 17], each pair of operators ( φ p , N p ) satisfies the canonicalcommutation relation in a subspace Ω p of the p -th Hilbert space L ( p − Z p ) as follows Ω p = n | f i ( p ) = ∞ X n p =0 f n p | n p i ( p ) : ∞ X n p =0 f n p = 0 o where, | n p i ( p ) is an eigenstate of N p corresponding to the eigenvalue n p . Thus, each ofthe phase operators φ p satisfies the canonical commutator over a dense subspace of theHilbert space h p φ p , X p ∈ prime ln D ( p ) i = i, in Ω p O p ′ = p H p ′ This is similar to the operator defined earlier, but without the sum restricted to a finiteprime p (along with the normalisation factor π ( p ) in the denominator). This is due to thefact that in this case, one gets a contribution from only one of the subspaces (one prime)at a time. In the next subsection we shall take a similar route to define another phaseoperator. Following [15] (see also [17]) we would like to discuss another construction of the phaseoperator ˆΦ conjugate to the number operator ˆ N such that ˆ N | n i = n | n i , n = 0 , , · · · , n ˆΦ = X m = n i | m ih n | m − n (24)13his is a ( n + 1) × ( n + 1) hermitian Toeplitz matrix [25–27]. When applied on a state | v i = P n v n | n i , we find that (cid:2) ˆΦ , ˆ N (cid:3) | v i = i | v i if and only if n X n =0 v n = 0 (25)Thus the commutator is valid in a codimension one subspace. For example, we couldchoose the v n s to be the nontrivial ( n +1) -th roots of unity. Toeplitz matrices and operatorshave a long history and have been studied extensively (see e.g., [27]). Although eigenvalues k ≤ k ≤ · · · ≤ k n and the corresponding eigenvectors | k m i ( k = 0 , , · · · , n ) of thematrix above exist, one cannot write them explicitly. Moreover, by Szegö’s theorems,the spectrum is bounded by π as n → ∞ (so that the matrix size goes to infinity) andthe eigenvalues are distributed uniformly and symmetrically around zero, as one can alsocheck numerically for small values of n .Coming back to the problem of our interest, in which the Hamiltonian is H = P p ln p N ( p ) , where N ( p ) is the number operator at the p -th site, which in turn can beexpressed in terms of the generalised Vladimirov derivative. For a natural number n ∈ N ,we use the prime factorisation to associate a vector in ⊗ p L ( p − Z p ) as n = Y p p n ( p ) ←→ | n i = ⊗ p | n ( p ) i (26)using the wavelet basis. We emphasise that only a finite number of entries in the infinitecomponent vector are non-zero integers. Clearly | n i is an eigenvector of HH | n i = X p n ( p ) ln p | n i = ln n | n i Moreover, these states are orthonormal h n i | n j i = Q p h n i ( p ) | n j ( p ) i = Q p δ n i ( p ) n i ( p ) = δ n i n j .When restricted to a fixed value of p , the following definition for the phase operator i ln p X n ( p ) a = n ( p ) b | n ( p ) a i h n ( p ) b | ( n ( p ) a − n ( p ) b ) on L ( p − Z p ) is natural. This is a Toeplitz matrix, therefore, it has eigenvectors | k ( p ) i .Let us define the phase operator on the full space ⊗ p L ( p − Z p ) schematically to be of theform Φ tot ∼ X n a = n b i | n a ih n b | ln n a − ln n b = X not all n ( p ) a = n ( p ) b i (cid:16) ⊗ p a | n ( p a ) i i (cid:17) (cid:16) ⊗ p b h n ( p b ) b | (cid:17)P p ( n ( p ) a − n ( p ) b ) ln p We need to specify the limits of sums over the integers in the above. However, beforewe undertake that exercise, we would like to check if H and Φ tot could be a canonically Prime factorisation also plays an important role in the arithmetic gas models [4–7]. finite linear combination of the form | v i = P n v n | n i ,in which we further require the coefficients to factorise as v n ≡ v ( n (2) ,n (3) , ··· ) = Q p v n ( p ) .One can compute the commutator and verify that on such a state (cid:2) Φ tot , H (cid:3) | v i = i | v i if and only if X n v n = 0 (27)where the upper limit of the sum is the maximum integer n max that appear in the definitionof the vector | v i . Consider all the vectors | n i that appear in the linear combination indefining | v i on which we want to check for the commutator, and the prime factorisationsof the corresponding integers n . Let the maximum max p { n ( p ) } of these be n ∈ N . Thereis also a highest prime p , i.e., above which all n ( p> p ) = 0 in the factorisations. We cannow make the proposal for the phase operator more precise. It is Φ tot = n X n ( p ) a , n ( p ) b =0 not all n ( p ) a = n ( p ) b i (cid:16) ⊗ p a ≤ p (cid:12)(cid:12) n ( p a ) i (cid:11)(cid:17) (cid:16) ⊗ p b ≤ p (cid:10) n ( p b ) b (cid:12)(cid:12)(cid:17)P p ≤ p ( n ( p ) a − n ( p ) b ) ln p for n ( p ) a , n ( p ) b ≤ n O p> p (cid:12)(cid:12) n ( p ) = 0 (cid:11)(cid:10) n ( p ) = 0 (cid:12)(cid:12) otherwise (28)and acts on a space spanned by vectors of the form | k , p i = (cid:0) ⊗ p ≤ p | k ( p ) i (cid:1) ⊗ (cid:0) ⊗ p ′ > p | n ( p ′ ) = 0 i (cid:1) where at least one k ( p ) = 0 for a prime p ≤ p and for p > p , we have chosen the ‘vacuum’state in the number representation. In the limit p → ∞ (even for finite values of n ), weexpect this to be a well defined Toeplitz operator on ⊗ p L ( p − Z p ) . However, we are notable to offer a rigorous mathematical proof of this assertion.The phase operator cannot be defined uniquely, it is ambiguous upto a similaritytransform [15]. Given a phase operator Φ , for example, as defined in Eq. (24), let usconsider the operator Φ β = e − βN Φ e βN related by a similarity transform labelled by aparameter β . This is would have been a trivial statement had the commutator Eq. (25)been true in the full vector space, however, as we have seen this relation holds in a subspaceof codimension one. It is straightforward to check that the condition that restricts to thesubspace is modified to P n e βn v n = 0 for Φ β to be conjugate to N . We may choose v n = e πim n/ ( n +1) and β = 2 πim / ( n + 1) with m + m = 0 (mod n + 1) . This conditionis identical to the vanishing of the partition function Z = Tr e − βH for the Hamiltonian H = − N at these special values of β .Now consider Φ tot ,β = e − βH Φ tot e βH , the similarity transformation of Eq. (28). Themodified condition that defines the subspace is Q p P n ( p ) e βn ( p ) ln p v n ( p ) = 0 . If we choosethe coefficient v n ( p ) = χ ( p ) n ( p ) , where χ ( p ) is a Dirichlet character (see Eq. (30)), the15ubspace is defined by the vanishing of p Y p =2 n X n ( p ) =0 (cid:0) χ ( p ) p β (cid:1) n ( p ) = p Y p =2 − χ ( p n +1 ) p β ( n +1) − χ ( p ) p β (29)which, in the limit p → ∞ is a ratio of Riemann zeta or Dirichlet L -functions, dependingwhether the character is trivial or not, as in Eqs. (19) and (38), respectively. Thus thesubspace in which the phase operator Eq. (28), or its similarity transform, is canonicallyconjugate to the Hamiltonian is defined by the vanishing of the Riemann zeta function(at special values of the inverse temperature β = 0 ). We have previously encountered thisin Eq. (19) with the aggregate phase operator defined in Section 3.1. As we see, differentchoices for the coefficients relate to the vanishing of Dirichlet L -functions, to which weshall now turn our attention. L -functions The Riemann zeta function belongs to a family of functions, called the Dirichlet L -functions, that are defined as the analytic continuation of the Dirichlet series L ( s, χ ) = ∞ X n =1 χ ( n ) n s = Y p ∈ primes − χ ( p ) p − s , Re( s ) > (30)to the complex s -plane. In the above, χ ( n ) , called the Dirichlet character, is a homomor-phism from the multiplicative group G ( k ) = ( Z /k Z ) ∗ of invertible elements of Z /k Z to C ∗ , which is then extended as a character for all Z by setting χ ( m ) = 0 for all m whichare zero (mod k ) [28]. A Dirichlet character so defined satisfy the following properties1. For all m , m ∈ Z , χ ( m m ) = χ ( m ) χ ( m ) χ ( m ) = 0 if and only if m is relatively prime to k χ ( m ) = χ ( m ) if m ≡ m (mod k ) Therefore, χ is a multiplicative character, defined modulo k , on the set of integers. It isthis multiplicative property that justifies the sum to be written as an infinite product inEq. (30).There is a trivial character that assigns the value 1 to all integers, including 0. (Thismay be taken to correspond to k = 1 .) The Riemann zeta function corresponds to thechoice of the trivial character. In all other cases, only those integers (respectively, primes),the Dirichlet characters of which are not zero, contribute to the sum (respectively, theproduct). This of course depends on the periodicity k of the character. Therefore, theproduct restricts to primes that do not divide k Y p − χ k ( p ) p − s = Y ( p,k )=1 − χ k ( p ) p − s = Y ( p,k )=1 ∞ X n p =0 χ k ( p ) n p p − n p s inverse χ − by restricting to the relevantset of primes. For these primes p ∤ k , the Dirichlet character satisfies χ − k ( p ) = χ ∗ k ( p ) .(Formally, for the others, we may take χ as well as χ − to be zero.)Everything we discussed in the context of the Riemann zeta function in Section 3,including all the caveats, apply to the Dirichlet L -functions, with obvious modificationsat appropriate places. The role of the generalised Vladimidrov derivative, acting on com-plex valued functions on the p -adic numbers Q p is played by the generalised Vladimirovderivative twisted by the character χ [20], be denoted by D ( p ) x . The Kozyrev wavelets areeigenfunctions of these operators for all χ . The eigenvalues, however, are different andinvolve the Dirichlet character as follows D ( p ) x ψ ( p )1 − n,m,j ( ξ ) = χ k ( p n ) p n ψ ( p )1 − n,m,j ( ξ ) (31)We refer to [20] for details of the construction and other properties of these operators.The above equation and Eq. (9) lead to the conclusion that D and D x are simulta-neously diagonalisable, hence the Kozyrev wavelets are also eigenfunctions of the unitary operator U x = D x D − U ( p ) x ψ ( p )1 − n,m,j ( ξ ) = D ( p ) x D − p ) ψ ( p )1 − n,m,j ( ξ ) = χ k ( p n ) ψ ( p )1 − n,m,j ( ξ ) (32)We can define its inverse U − p ) x = U † ( p ) x = D ( p ) D − p ) x ∗ for those k which do not contain p inits factorisation (otherwise it is the identity operator). Conversely, when we consider allprimes, for a given k , we need to restricted to the set of primes that do not divide k , i.e.,with the formal extension of the inverse given after ?? .As in the case of the Riemann zeta function, we can combine all the prime factors towrite L ( s, χ k ) as a trace L ( s, χ ) = X n =( n (2) ,n (3) , ··· ) (cid:10) n (cid:12)(cid:12) U x e − s ln D (cid:12)(cid:12) n (cid:11) where U x = ⊗ p U ( p ) x . If we want to interpret this as the partition function of a statisticalmechanical model, the Hamiltonian is such that e − βH x ←→ U x e − s ln D = D − D x D − s = D x D − s − which reduces to e − βH ∼ D − s corresponding to the Riemann zeta function in Section 3,upto a phase. Now since a non-zero χ k ( p ) = e iω p is a root of unity, we can define a newphase state | φ ( x ) k p i = 1 √ n + 1 n X n p =0 e − in p ( φ kp ln p + ω p ) | n p i (33)which provide an orthonormal set (cid:10) φ ( x ) k ′ p (cid:12)(cid:12) φ ( x ) k p (cid:11) = δ k p ,k ′ p This is done by truncating the spectrum to relate with the previous case. ˆ φ ( p ) x = X k p (cid:18) φ k p + ω p ln p (cid:19) (cid:12)(cid:12) φ ( x ) k p (cid:11)(cid:10) φ ( x ) k p (cid:12)(cid:12) ≡ X k p φ ( x ) k p (cid:12)(cid:12) φ ( x ) k p (cid:11)(cid:10) φ ( x ) k p (cid:12)(cid:12) (34)using the eigenvalues and eigenstates as before.Now we define the operator U ( p ) e β ln pN p such that U ( p ) e β ln pN p | n p i = e in p ω p e βn p ln p | n p i (35)It follows that ˆ φ ( p ) x U ( p ) e β ln p N p = U ( p ) e β ln p N p X k p (cid:18) φ ( x ) k p + ω p ln p − iβ (cid:19) (cid:12)(cid:12)(cid:12) φ ( x ) k p + ω p ln p − iβ ED φ ( x ) k p + ω p ln p − iβ (cid:12)(cid:12)(cid:12) + U ( p ) e β ln p N p (cid:18) iβ − ω p ln p (cid:19) X k p (cid:12)(cid:12)(cid:12) φ ( x ) k p + ω p ln p − iβ ED φ ( x ) k p + ω p ln p − iβ (cid:12)(cid:12)(cid:12) This relation can be obtained by the same method used in the earlier sections. If iβ takesany of the values πk p ( n +1) ln p + ω p ln p then in the first term above, one gets the phase operatorEq. (34), since φ k p is defined modulo π ln p . Hence, as in the case of the Riemann zetafunction, h ˆ φ ( p ) x , U ( p ) x e β ln p N p i = (cid:18) iβ − ω p ln p (cid:19) U ( p ) x e β ln p N p (36)The definition of the (exponential of the) resolvent is completely analogous to the case ofthe Riemann zeta function Eq. (22) — one only needs to substitute ˆ φ p → ˆ φ ( p ) x , resultingin the trace p X p =2 X kp exactly one φkp =0 − e i ( φ kp + ωp ln p ) − iφ + X kp at least two φkp =0 − e iωp ln p − iφ ! in place of Eq. (23). Once again the poles (apart from that at φ = 0 ) coincide with thezeroes of the partition function, which is Z ( β ) = Tr U x , p exp (cid:16) β p X p =2 ln p N p (cid:17)! = p Y p =2 − χ n +1 ( p ) p β ( n +1) − χ ( p ) p β (37)where we have used the fact that χ ( p n +1 )( p ) = χ n +1 ( p ) is again a character with the sameperiodicity. In the thermodynamic limit p → ∞ we get the following ratio of the Dirichlet L -functions Z ( β ) = L ( − β, χ k ) L ( − ( n + 1) β, χ n +1 k ) (38) The unitary opeartor U x , p is the product of the corresponding operators at all sites U x , p = p Y p =1 U ( p ) x .
18n the special case where n + 1 is the Euler totient function ϕ ( k ) or its integer multiple, χ ( p ϕ ( k ) ) = ( χ ( p )) ϕ ( k ) = χ k, ( p ) , the principal character, which is 1 if ( p, k ) = 1 and 0otherwise. Except for the trivial zeroes, the script of the discussions above is very similarto what we argued for the Riemann zeta function.In summary, we have proposed to view the Riemann zeta and the Dirichlet L -functionsas the partition functions (upto multiplication by a function that plays no essential role)of quantum spins in magnetic fields, the values of which depend on the site. We haveargued how to make sense of the phase operator (upto a similarity transformation). Thezeroes of the partition function coincide with the poles of the resolvent function of theexponential of the aggregate or total phase operators, as discussed in Sections 3.1 and 4.A different approach to the phase operator was discussed in Section 3.2. Its relation tothe partition function, via similarity transforms, seems to relate the zeta function and the L -functions in the same framework. Acknowledgements:
We thank Surajit Sarkar for collaboration at initial stages of thiswork. It is a pleasure to acknowledge useful discussions with Rajendra Bhatia and VedPrakash Gupta. We would like to thank Toni Bourama and Wilson Zùñiga-Galindo forthe invitation to write this article.
References [1] H. Montgomery, “The pair correlation of zeros of the zeta function,”
Analytic numbertheory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo.,1972) , pp. 181–193, 1973.[2] B. Hayes, “Computing science: the spectrum of Riemannium,”
American Scientist ,vol. 91, no. 4, pp. 296–300, 2003.[3] A. Odlyzko, “ The -nd zero of the Riemann zeta function,” in Dynamical, spectral,and arithmetic zeta functions (San Antonio, TX, 1999), Contemp. Math. , pp. 139–144, 2001.[4] D. Spector, “Supersymmetry and the Möbius inversion function,” Comm. Math.Phys. , vol. 127, pp. 239–252, 1990.[5] B. Julia,
Statistical theory of numbers . in Number Theory and Physics, J. Luck,P. Moussa, and M. Waldschmidt (Eds.), Springer Proceedings in Physics, Springer,1990.[6] B. Julia, “Thermodynamic limit in number theory: Riemann-Beurling gases,”
PhysicaA: Statistical Mechanics and its Applications , vol. 203, no. 3, pp. 425 – 436, 1994.[7] I. Bakas and M. Bowick, “Curiosities of arithmetic gases,”
Journal of MathematicalPhysics , vol. 32, pp. 1881–1884, 1991. 198] A. Knauf, “ Phases of the number-theoretic spin chain,”
J. Stat. Phys. , vol. 73,pp. 423–431, 1993.[9] A. Knauf, “ The number-theoretical spin chain and the Riemann zeroes,”
Commun.Math. Phys. , vol. 196, pp. 703–731, 1998.[10] D. Schumayer and D. Hutchinson, “ Physics of the Riemann hypothesis,”
Rev. Mod.Phys. , vol. 83, pp. 307–330, 2011, 1101.3116 [math-ph].[11] C. Itzykson and J.-M. Drouffe,
Statistical field theory: vol. 1, From Brownian motionto renormalization and lattice gauge theory . Cambridge Monographs on MathematicalPhysics, Cambridge University Press, 1991.[12] L. Susskind and J. Glogower, “ Quantum mechanical phase and time operator,”
Physics , vol. 1, pp. 49–61, 1964.[13] P. Carruthers and M. Nieto, “ Phase and angle variables in quantum mechanics,”
Rev.Mod. Phys. , vol. 40, pp. 411–440, 1968.[14] J. Garrison and J. Wong, “ Canonically conjugate pairs, uncertainty relations andphase operators,”
J. Math. Phys. , vol. 11, pp. 2242–2249, 1970.[15] A. Galindo, “ Phase and number,”
Lett. Math. Phys. , vol. 8, pp. 495–500, 1984.[16] D. Pegg and S. Barnett, “Unitary phase operator in quantum mechanics,”
Europhys.Lett. , vol. 6, pp. 483–487, 1988.[17] P. Busch, M. Grabowski, and P. Lahti,
Operational quantum physics , vol. 31 of
Lecture Notes in Physics . Springer, 1995.[18] X. Ma and W. Rhodes, “ Quantum phase operator and phase states,” arXiv e-print ,2015, arXiv:1511.02847 [quant-ph].[19] A. Chattopadhyay, P. Dutta, S. Dutta, and D. Ghoshal, “ Matrix Model for Riemannzeta via its local factors,”
Nucl. Phys.
B954 , p. 114996, 2020, 1807.07342.[20] P. Dutta and D. Ghoshal, “ Pseudodifferential operators on Q p and L -Series,” 2020,arXiv:2003.00901.[21] P. Busch, M. Grabowski, and P. Lahti, Operational quantum physics . Lecture Notesin Physics, Springer, 2013.[22] S. Kozyrev, “ Wavelet theory as p -adic spectral analysis,” Izv. Math. , vol. 66, no. 2,p. 367—376, 2002, arXiv:math-ph/0012019.[23] P. Dutta, D. Ghoshal, and A. Lala, “ Enhanced symmetry of the p -adic wavelets,” Phys. Lett. , vol. B783, pp. 421–427, 2018, 1804.00958.[24] A. Khrennikov, S. Kozyrev, and W. Zúñiga-Galindo,
Ultrametric pseudodifferentialequations and applications . Encyclopedia of Mathematics and its Applications, Cam-bridge University Press, 2018. 2025] R. Gray, “ Toeplitz and circulant matrices: a review,”
Foundations and Trends inCommunications and Information Theory , vol. 2, pp. 153–239, 2006.[26] H. Widom,
Toeplitz matrices . in Studies in real and complex analysis, I. HirschmanJr. (Ed.), The Mathematical Association of America, 1990.[27] N. Nikolski,
Toeplitz matrices and operators . Cambridge Studies in Advanced Math-ematics, Cambridge University Press, 2020.[28] J.-P. Serre,