_{1}

By interpreting multifractal L-function zero alignment as a decoherence process, the Riemann hypothesis is demonstrated to imply the emergence of classical phase space at zero alignment. This provides a conception of emergent dynamics in which decoherence leads to classical system formation, and classical system trajectories are characterized by modular forms.

Throughout the twentieth century, a preoccupation of theoretical physics has been to identify the fundamental constitutents of matter and understand how they behave. This preoccupation has led to the construction and operation of increasingly larger particle colliders with which these constituents have been studied with greater and greater precision, and ultimately, to the discovery and validation of the Standard Model of particle physics. This model stands as a testament to the efforts of many people, and some might claim it constitutes a theory of everything once a consensus is reached on how to incorporate the gravitational force [

Notably, at the root of this claim, there lies a reductionist view of the natural world, born out of extensive agreement of atomic models with experiment, and the direct observation of atoms and elementary particle tracks with scanning tunneling microscopes and particle colliders. For some, this evidence is strong enough to conclude that the Standard Model of particle physics constitutes an understanding of all biology, and even consciousness in that it describes in principle all biochemical mechanisms at an atomic level. Of course, this point of view is not universal, since scientists studying natural phenomena whose features of interest are not explained by atom-scale models may draw different conclusions, and regard such claims about human understanding as scientific overreach.

Interestingly, despite the many successes of quantum physics, there are basic theoretical questions surrounding it that remain unresolved. For instance, there is no entirely satisfactory explanation for how a measured quantum system collapses into an observable state. Secondly, though often taken for granted, it is a feature of all closed quantum systems that they undergo unitary evolution in time, because the eigenvalues of the time evolution operator are complex numbers lying on a circle of unit radius. This time evolution operator is deter- mined by the interaction and kinetic energies of a configuration of particles in space, and its success as a descriptor of atomic physical systems provides the theoretical basis for reductionism.

Given this situation, the purpose of this paper is to apply number theory to investigate the possibility that non-unitary evolution is the prime mover driving physical change. Our investigation proceeds via the study of open quantum systems which exhibit non-unitary evolution in time. From a conventional perspective, this non-unitary evolution, known as decoherence, or state mixing, is a consequence of unitary evolution of the open quantum system and its en- vironment considered as a whole. However, in this paper we’ll present a different point of view, from which quantum unitary evolution emerges as a special limit of non-unitary evolution.

In terms of layout, Chapters 2 - 4 outline research interests that motivated this work. For example, understanding how the physics of open quantum systems may be relevant to the workings of biological systems intricately coupled to their environment is discussed. Switching modes, Chapter 5 introduces the theory of solitary waves, and Chapter 6 elaborates on this discussion, introducing tau functions, modular forms and L-functions. Using these ideas, Chapter 7 introduces an alignment process analogous to state mixing that leads to the emergence of quantum unitary evolution and classical phase space, and a conjecture is made about how this emergence relates to the Standard Model. Chapter 8 concludes with a summary of results, and explains why they are of scientific interest.

Classical physical theories such as electrodynamics describe physical systems as configurations of particles and fields. In these theories, a particle such as an electron or proton is idealized as a point in three dimensional space, as shown in

influence particle motion and particle motion influences fields are taken into account simultaneously.

Importantly, Maxwell’s equations are differential equations describing smooth evolution of particle and field configurations in time and space. Mathematically, this relies on the assumption that time and space dimensions are coordinatized by 4 real numbers

Interestingly, this issue is not particular to classical electrodynamics, but persists generally in classical and quantum mechanical descriptions of Nature, where we can similarly imagine the differential equations describing physical systems in time and space are approximations of underlying difference equations. This situation is not entirely satisfying, because it leaves us ignorant as to whether time and space are continuous, discrete, or better understood from a different point of view.

In classical mechanics, a point particle constrained to move in one dimension is described by its position and momentum at any given moment in time. That is, assuming its position and momentum are coordinatized by real numbers

Practically speaking, classical mechanics is well equipped to model closed physical systems, but not open systems. For instance, to usefully model cell division with classical mechanics, we are forced to somewhat arbitrarily partition the phase space of the cell and its environment together into separate cell and environmental phase spaces. That is, it is necessary to identify all of the particles playing a role in the cell’s division, and model this division as a process governed by interactions between these particles and some average environmental effect. Unfortunately, this description does not allow for unpredictable variations in temperature, pressure, or particle exchange between the cell interior and exterior, making precise modeling impossible. Moreover, in the case of cell division, these sources of imprecision are complicated by the extremely large number of particles involved in all phases of the process. This situation is illustrated in

Another interesting feature of classical mechanics is that system time evolution tends to be disordered. That is, classical system trajectories are generically chaotic, filling entire

Finally, as a technical point, we note that while classical physics describes the real time evolution of fields as well as particles, there is no clear choice of configuration space in which field configurations flow like there is for particles. That is, if we ask what the set of physically realizable electric field configurations across the Euclidean space

matical criteria these functions should satisfy, or how to define a probability measure on this function space as necessary to describe the statistical behavior of the field in a thermal environment.

In quantum mechanics, the Heisenberg uncertainty principle states that precise position and momentum coordinates of particles are not simultaneously specifiable. Consequently, the mathematical description of quantum particles is given in terms of position or momentum probability distribution functions, not points in phase space, and closed multi-particle systems are described by wave functions of position or momentum coordinates that evolve in time according to the dictate of a Hamiltonian energy operator rather than a Hamiltonian vector field. This operator defines real valued system energy levels, and unitary evolution of wave functions according to Schrodinger’s equation. A similar mathematical formalism describes the time evolution of quantum fields, though computations are typically performed via evaluation of Feynman diagrams rather than directly solving the Schrodinger equation. As in the case of classical mechanics, quantum mechanical modeling of biological systems with environ- mental interactions is awkward when system and environmental variables are difficult to distinguish.

One crucial difference between quantum and classical descriptions of physical systems is the effect of measurement on these systems. This difference stems from the description of quantum particles as wave functions spread out over all of position space, whereby a particle-like quality of these entities is only realized upon measurement with an experimental apparatus. The prototypical example of this is the observation of particle position on a detecting screen in Young’s double slit experiment, in which observation of classical particle-like behavior absent in the mathematical description of wave functions is referred to as wave function collapse. Intuitively, one expects collapse to be a consequence of the interaction of a quantum mechanical system with its measurement apparatus, as required to observe the system. For this reason, collapse and our experimental observation of particles is inherently related to the behavior of open quantum systems. Philosophically, this is important, because it leaves open the possibility that our classical notion of “particle” emerges from a description of open quantum systems in which this notion is not fundamental.

Turning to the study of open quantum systems, it is common to use density matrices instead of wave functions to describe system evolution, because this formalism can account for environmentally induced state transitions [

by averaging over environmental degrees of freedom. The upshot of this description is that most pure quantum states of the open system are unstable, and evolve into statistical mixtures of pointer states that are stable against further mixing [

As mentioned, in the commuting case, state mixing results in the off-diagonal decay of the system density matrix written in a pointer state basis. Furthermore, in the event the environment acts as a heat bath at thermal equilibrium, the diagonal weights of the density matrix evolve towards an equilibrium distribu- tion in which each pointer state is weighted by a Boltzmann factor. This process, known as relaxation, typically takes place on timescales much longer than dephasing.

Importantly, evolution of a system density matrix into a statistical mixture of pointer states is mathematically distinct from the projection of a density matrix into a pure quantum state that occurs with measurement of the system. This projection, known as collapse, has the effect of restoring a pure quantum state that can once again evolve into a mixed state upon environmental interaction. From a theoretical point of view, it is understood why measured quantum sys-

tems evolve into statistical mixtures of pointer states, because they are necessarily open to their environment. However, it is not clear how collapse into a single observable outcome occurs, and this absence of clarity lies at the heart of the measurement problem.

To obtain a classical approximation of a quantum system, the Wigner trans- form can be applied to the quantum system density matrix to construct a classical trajectory distribution on classical phase space. Depending on the mixed state represented by the density matrix, this construction is not always physically meaningful. However, in the event the density matrix represents a statistical mixture of spatially localized pointer states, applying the Wigner transform yields a time varying probability distribution on classical phase space that describes the likelihood of the system taking different classical trajectories. Conventionally, such spatially localized pointer states are called coherent states.

Remarkably, there are similarities between the theory of open quantum systems and number theory, whereby commuting

To explain how the alignment process described in Chapter 7 is driven, we turn to the theory of solitary waves (i.e. solitons). This theory is useful to us because differential equations describing the motion of solitons define geometric objects called Riemann surfaces

To begin explaining this, let’s take a look at the Korteweg de-Vries (KdV) equation, the prototypical soliton equation describing non-dispersive propaga- tion of waves in shallow water [

where

in differential operators

This Lax equation has time independent solutions of the form:

where

or:

using the relationship between

To better understand the relationship between the KdV equation and elliptic curves, let’s assume the differential operators

In this event, according to a result of Burchnall and Chaundy, the operators

of degree 3 in

More generally, we can construct soliton equations:

solved by functions

in which the differential operators

satisfying:

Similarly, fixing

satisfying:

Because conjugation of an operator does not change its eigenvalues, the eigenvalues of the operators

Assuming the differential operators

curve. Moreover, since there are

whose

whose solution

To introduce these ideas, let’s imagine that a state mixing process takes place in the Q-analog limit

defines a differentiation operator in

More specifically, upon substituting

where

Locally,

Remarkably, these branches resemble centromeres aligning chromosomes at metaphase, as shown in

This chapter introduces tao functions, explaining their relationship to soliton equations, theta functions, and modular forms. It also provides a brief introduc- tion to L-functions and the Riemann hypothesis, as necessary for understanding the discussion in Chapter 7. Note that tao functions are more commonly known as tau functions, but the name tao, meaning great waves, has been adopted here to avoid confusion with the modular parameter

Tao functions generate solutions to soliton equations. For example, a tao function

generates solutions to KdV Equation (3) via the relation:

For example, the Jacobi theta function appearing in Equation (9) is an example of a tao function solving Equation (23).

More generally, soliton equations of type (12) with

as:

where

Tao functions can also be defined when

whenever they solve the bilinear KP equation:

Once again, Riemann theta functions provide viable examples of tao func- tions.

Modular forms are functions

for some discrete subgroup

Demonstrating that the number of independent modular forms increases with weight

Other examples of weight

in which

This modular function satisfies a polynomial equation whose coefficients depend on the j-invariant

In physics, modular functions arise as renormalization flow limits of Ising model partition functions [

Summing over all possible spin configurations, this Ising energy generates a quantum partition function

that depends on the parameters

Assuming the spins

where

The renormalization transformation associated with this decimation and rescaling is:

and this transformation gives rise to regular and chaotic flows below and above the curve

Formally, under repeated iteration of renormalization transformation (35), the partition function

Artin L-functions

in that they are expressible as infinite products over primes. Technically, we can associate an L-function with each representation

where

Interestingly, it is conjectured that all non-trivial zeros of Artin L-functions lie along the critical line

known as the generalized Riemann hypothesis, has close ties with physics. For example, it has been proposed that alignment of the Riemann zeros is related to the Hermiticity of quantum operators and/or the alignment of Yang-Lee zeros of Ising model partition functions [

In this chapter, our goal is to explain how the Riemann hypothesis is related to the emergence of classical phase space, and how this emergence is driven. To this end,

graphically projected onto the surface of a Riemann sphere. Conjecturally, these critically aligned zeros act as attractors for a multifractal L-function zero flow carrying the zeros of

To understand the relationship between zero flow and state mixing, let’s assume the automorphic waveforms

on the adelic group

To relate multifractal zero alignment to the emergence of classical phase space, we’d like to associate zero flows with geometric objects

the Galois representation

acted on by the discrete group

In the special case

Geometrically, we can understand classical system formation in

With this in mind, let’s consider a variant of twistor theory in which twistors are replaced by continuous paths in

of

Visually, we can imagine system formation in

dence in a plane

Because the particle trajectory in

are Gaussian hypergeometric functions defining periods of classical rotational motion [

Intuitively, this conjecture is motivated by noting hypergeometric tao functions are combinatorial generating functions of signed Hurwitz numbers [

To understand this in greater detail, let’s imagine Equation (18) is integrated in the complex

into matrices

and Equation (47) is the Riemann-Hilbert factorization of

solving KZ differential equations, whose ratio is a unitary character of the Virasoro algebra [

Algebraically, the determinant of Riemann-Hilbert factorization (47) is a relation between scattering amplitudes in the Hopf algebra of Feynman diagrams

Interestingly, there are cases in which the the aforementioned relation between scattering amplitudes is a

satisfied by

As an example, let’s assume

generate a modular function field of degree 3 at

and the ratio:

is a cyclotomic unit of degree 3 in

is conjectured to have a continued fraction expansion for an appropriate choice of the partition function

Blending ideas from math and physics, this paper suggests state mixing is the fundamental process underlying the time evolution of physical systems. Formally, this is achieved by replacing quantum density matrices with multifrac- tal L-functions

Physically, the results of this paper are of interest because they highlight a connection between open quantum systems and number theory. Specifically, commuting system and environmental interaction operators

Mathematically, the drive towards multifractal zero alignment is explained using the theory of solitons. This is done by identifying the root space

Outside the realm of pure science, the results of this paper may also have real world applications. For example, as described, unitary mixing instills emergent classical systems with a balance between ordered and chaotic behavior that may be relevant to understanding the presence of self organized criticality in Nature [

Brox, D. (2017) The Riemann Hypothesis and Emergent Phase Space. Journal of Modern Physics, 8, 459-482. https://doi.org/10.4236/jmp.2017.84030