# Interacting Bose and Fermi gases in low dimensions and the Riemann hypothesis

###### Abstract

We apply the S-matrix based finite temperature formalism to non-relativistic Bose and Fermi gases in and dimensions. For the dimensional case in the constant scattering length approximation, the free energy is given in terms of Roger’s dilogarithm in a way analagous to the thermodynamic Bethe ansatz for the relativistic dimensional case. The fermionic case with a quasi-periodic 2-body potential could provide a physical framework for understanding the Riemann hypothesis.

## I Introduction

Quantum gases at finite temperature and density have very wide applications in nearly all areas of physics, ranging from black body radiation, Fermi liquids, Bose-Einstein condensation, and equations of state in cosmology. If one knows the zero temperature dynamics, i.e. the complete spectrum of the hamiltonian, then the quantum statistical mechanics just requires an additional statistical summation , so that in principle zero temperature dynamics and quantum statistical sums are decoupled. This decoupling is also in principle clear from an intuitive picture of a gas as a finite density of particles that are subject to scattering. In practice, the complete non-perturbative spectrum of is unknown so one must resort to perturbative methods, such as the Matsubara approach, that typically entangle the zero temperature dynamics from the quantum statistical mechanics.

In AndreSthermo an alternative approach to finite temperature quantum field theory was developed that achieves this decoupling of zero temperature dynamics and quantum statistical summation. The dynamical variables are the occupation numbers, or filling fractions, of the gas, and the zero temperature data of the underlying theory are just the S-matrix scattering amplitudes. Our construction was modeled after Yang and Yang’s thermodynamic Bethe ansatz (TBA)YangYang . The TBA is very specific to integrable theories in one spacial dimension. Nevertheless, our derivation AndreSthermo is carried out in any spacial dimension and doesn’t assume integrability; therefore it is in general an approximate method, though it can be systematically improved. An important ingredient of our construction is the work of Dashen, Ma, and BernsteinMa which explains how to formulate quantum statistical mechanics in terms of the S-matrix. Ideas of Lee and Yang were also instrumentalLeeYang , even though the latter work is not based on the S-matrix.

Our previous workAndreSthermo was mainly devoted to developing the formalism in a general, model independent way for both non-relativitistic and relativistic theories. Though the derivation was somewhat involved, the final result is straightforward to implement and is summarized by the two formulas (23, 24). In this paper we apply the method to non-relativistic Bose and Fermi gases with special attention paid to the low dimensional cases of . The main approximation we make is to consider only the contributions to the free energy that come from 2-particle to 2-particle scattering. On physical grounds these are expected to be the most important if the gas is not too dense. Even in this approximation the problem is non-trivial since one needs to resum all the 2-body interactions self-consistently. This is accomplished by an integral equation for the filling fractions that is analagous to the TBA equations.

In the next section we first review the main results in AndreSthermo . In section III we describe the kind of second-quantized hamiltonians that are the subject of this paper. In section IV we consider bosons with a function two-body potential. In section V we turn to fermionic gases in the same approximation and derive formulas that can be used to study the effect of interactions on the Fermi energy. Since the lowest order contribution corresponds to a constant scattering length approximation, the results we obtain in sections IV and V are already known, and are included here mainly for illustrative purposes.

In section IV we study the case which has some remarkable features. The integral equation which determines the filling fraction becomes algebraic. The formulas for the free energy are essentially identical to those for the conformally invariant limit of relativistic theories in one lower dimension ZamoTBA and are given in terms of Roger’s dilogarithm. In making this analogy we define a “central charge” as the coefficient in the free energy, which in the relativistic case in is the Virasoro central charge. As in the relativistic case, for certain special values of the coupling, can be a rational number, and some examples based on the golden mean are presented. We also extend the formalism to many species of mixed bosonic and fermionic statistics.

As will be clear in the sequel, the polylogarithmic and Riemann functions play a central role in this work. This provided us with a new opportunity to understand Riemann’s hypothesisEdwards . Riemann’s function for is defined as

(1) |

It has a simple pole at ; as we will describe this pole is a manifestation of the impossibility of Bose-Einstein condensation in . The function can be analytically continued to the whole complex plane. It has trivial zeros at where is a non-zero positive integer. The function has a duality relation that relates to (eq. (109) below) so that its non-trivial zeros are in the critical strip and symmetric around . The Riemann hypothesis (RH) is the statement that the only non-trivial zeros of are at . This hypothesis is very important in number theory since, as shown by Riemann, the distribution of prime numbers is intimately related to the location of the Riemann zeros. It has already been proven by Hadamard and de la Vallée-Poussin that there are no zeros with , which is equivalent to proving the Prime Number TheoremEdwards .

There have been a number of approaches to understanding
the Riemann hypothesis based on physics^{1}^{1}1For a comprehensive
list see M. Watkins at
http://secamlocal.ex.ac.uk/ mwatkins/zeta/physics.htm.. Here we mention
a few of them mainly to contrast with the approach presented here.
Some approaches are inspired by the Polya-Hilbert conjecture
which supposes that the imaginary part of the zeros
corresponds to eigenenergies of
some unknown quantum mechanical hamiltonian.
The well-studied statistical properties
of the zeros, in particular the Montgomery-Odlyzko law
and its relation to the gaussian unitary ensemble of random
matricesMontgomery ; Odlyzko , led
Berry to propose a quantum chaos interpretation
of the oscillatory part of Riemann’s counting formula
for the zerosBerry . This is still within the context of
quantum mechanics at zero temperature.
Connes has proposed the complementary picture
that the
zeros correspond to absorption linesConnes .
Sierra has recently proposed a consistent quantization of
the Berry-Keating hamiltonian based on the Russian doll model
of superconductivity, which possesses a cyclic renormalization
group flowSierra .

The fundamental duality relation relating to
can be understood as a consequence of
a special modular transformation for the quantum statistical
mechanics of free, relativistic, massless particles in
spacetime dimensions. This is explained in
the appendix, since it is tangential to the main development of
this paper.
In section VII Riemann’s
function on the critical strip
is related to the quantum statistical mechanics
of non-relativistic,
interacting fermionic gases in with a quasi-periodic
2-body potential which depends on . This is thus
a many-body problem at finite temperature. It is perhaps not
completely unanticipated that the RH could have a resolution in
the present context, since one of the very first places in
physics where appeared was in Planck’s work on black body
radiation, which is widely acknowledged as the birth of quantum
mechanicsPlanck .
The black body theory is
the bosonic quantum statistical mechanics of a gas of free relativistic
photons^{2}^{2}2It is not clear
from Planck’s paper whether he was aware he
was dealing with Riemann’s function.
He simply writes .
In this case what appears is .
As we describe here, in order to get into the critical strip
one needs non-relativistic fermions interacting with a quasi-periodic
potential. The fermionic nature renders the required integrals
convergent and is also needed for a well-defined hamiltonian.
This quasi-periodicity implies our approach is closest to
Sierra’sSierra ,
since there also a periodicity was
important, but in the renormalization group at zero temperature.
Our approach is thus essentially different just because
the physical context is different, however
our interest in the RH actually stemmed from the workRD
on cyclic renormalization group flows in relativistic
systems and the observation that the finite temperature
behavior was in part characterized by ,
where is the period of the renormalization group flowRD2 .
We now understand that one needs to consider non-relativistic
models to get in a natural way^{3}^{3}3There
are also some convergence problems with the usual TBA, which
are avoided in the present work..

The understanding of the RH that emerges from the present context can be summarized as follows. When , there exists a two-body potential in position space that leads to a well-defined quasi-perodic kernel in momentum space that is essentially a combination of Fourier transforms of the potential. The condition comes about naturally as the condition for the convergence of the appropriate Fourier transform. The quantum statistical mechanics based on this kernel gives corrections to the pressure of the gas that are determined by a transcendental equation involving the polylogarithm . To obtain this, one must work with the momentum-dependent kernel, i.e. the constant scattering length approximation made in earlier sections vanishes. If then there would exist solutions with vanishing corrections to the pressure. Therefore the RH would follow from the simple physical property that non-zero interactions necessarily modify the pressure. We comment on the meaning of the actual zeros in section VII.

## Ii Free energy as a dynamical functional of filling fraction

### ii.1 Generalities

The free energy density (per volume) is defined as

(2) |

where are the inverse temperature and chemical potential, is the d-dimensional spacial volume, and and are the hamiltonian and particle number operator. Since is an extensive quantity, i.e. proportional to the volume, the pressure of the gas is minus the free energy density:

(3) |

For most of this paper, we assume there is one species of bosonic () or fermionic () particle. Given , one can compute the thermally averaged number density :

(4) |

where is the d-dimensional momentum. The dimensionless quantities are called the filling fractions or occupation numbers.

One can express as a functional of in a meaningful way with a Legendre transformation. Define

(5) |

Treating and as independent variables, then using eq. (4) one has that which implies it can be expressed only in terms of and satisfies .

Inverting the above construction shows that there exists a functional

(6) |

which satisfies eq. (4) and is a stationary point with respect to :

(7) |

The above stationary condition is to be viewed as determining as a function of . The physical free energy is then evaluated at the solution to the above equation. We will refer to eq. (7) as the saddle point equation since it is suggestive of a saddle point approximation to a functional integral:

(8) |

In a free theory, the eigenstates of are multi-particle Fock space states . Let denote the one-particle energy as a function of momentum . In this paper the theory is assumed to be non-relativistic with

(9) |

where is the mass of the particle. It is well-known that the trace over the multi-particle Fock space gives

(10) |

From the definition eq. (4) one finds the filling fractions:

(11) |

In order to find the functional one first computes from eq. (5) and eliminates to express it in terms of using eq. (11). One finds

(12) |

One can then easily verify that has the solution and plugging this back into eq. (12) gives the correct result eq. (10) for . In the sequel, it will be convenient to trade the chemical potential variable for the variable :

(13) |

There is another way to view the above construction which involves the entropy. Write eq. (12) as

(14) |

where is the first term in eq. (12), which is the energy density, and is the remaining term in brackets. One can show by a standard counting argument, which involves the statistics of the particles, that represents the entropy density of a gas of particles. (See for instance Landau .)

Let us now include interactions by writing

(15) |

where is given in eq. (13) and we define as the “potential” which depends on and incorporates interactions:

(16) |

Given , is determined by the saddle point equation . It is convenient to define a pseudo-energy as the following parameterization of :

(17) |

Then the saddle point equation and free energy density take the form:

(18) | |||||

(19) |

### ii.2 Two-body approximation

It was shown in AndreSthermo how to express in terms of S-matrix scattering amplitudes. In general there are terms involving the forward scattering of numbers of particles for . The most important is the two-body contribution. It takes the form

(20) |

where we have defined the convolution

(21) |

and the kernel is given by the 2-particle to 2-particle forward scattering amplitude:

(22) |

How to compute the kernel from the hamiltonian is described below.

In terms of the pseudo-energy, the saddle point equation and free energy take the following forms:

(23) | |||||

(24) |

## Iii Second quantized hamiltonians

In this paper we consider second quantized hamiltonians of the form:

(25) |

where is the 2-body potential. The field satisfies the canonical commutation relations

(26) |

where again corresponds to bosons/fermions. Expanding the field in terms of annihilation operators,

(27) |

this leads to the canonical (anti-) commutation relations:

(28) |

The free Hilbert space is thus a bosonic or fermionic Fock space with momentum eigenstates normalized as: .

In this paper we only consider the lowest order contribution to the kernel . Let denote the interacting part of which depends on the 2-body potential . Then to lowest order,

(29) |

where is the scattering amplitude for the asymptotic states . Therefore to lowest order the kernel is given by

(30) |

## Iv Hard core boson model revisited

In this section we consider bosonic particles with a delta-function two-body potential:

(31) |

The model has been solved exactly in by Lieb and LinigerLieb , and the thermodynamic Bethe ansatz was first discovered in the context of this modelYangYang .

Using , one finds

(32) |

Positive corresponds to repulsive interactions. Since the kernel is constant in this approximation, it is equivalent to the constant scattering length approximation made in the literature; hence we do not obtain any new results for the hard core Bose gas here. We include this section and the next mainly for illustrative purposes and as a warm-up to the sequel.

The coupling constant has units of . It can be expressed in terms of a physical scattering length as follows. To first order in perturbation theory the differential cross-section in the center of mass is:

(33) |

where is the magnitude of for one of the incoming particles. Since a cross section has dimensions of , we define such that the cross section is when the wavelength of the particle is :

(34) |

This leads us to make the definition:

(35) |

We carry out our analysis for arbitrary spacial dimension . Using where is the standard -function, rotationally invariant integrals over momenta can be traded for integrals over :

(36) |

For a constant kernel , and , the solution to the integral equation eq. (23) takes the simple form:

(37) |

where is independent of . To determine one needs the integral:

(38) |

The function is the standard polylogarithm, defined as the appropriate analytic continuation of

(39) |

For fixed , the function has a branch point at with a cut along , and the above integral is not valid at . We will later need the fact that when , the above integral is only convergent when and is given in terms of Riemann’s function:

(40) |

Equivalently,

(41) |

Eq. (23) then leads to the following equation satisfied by :

(42) |

where we have defined the fugacities

(43) |

and a renormalized thermal coupling and thermal wavelength :

(44) |

To simplify subsequent expressions, we henceforth set the mass , unless otherwise stated. This leads us to define:

(45) |

The equation (42) is a transcendental equation that determines as a function of , and the coupling . Given the solution of this equation, using eq. (24) the density can be expressed as

(46) |

For one can integrate by parts and obtain the following expressions for the free energy

(47) |

Bose-Einstein condensation can be described rather generally as follows. The property that an extensive, i.e. proportional to the volume, number of particles occupy the ground state with implies that the filling fraction diverges at some critical chemical potential :

(48) |

Given , one can speak of a critical density :

(49) |

One also has a critical temperature defined as

(50) |

where is the physical density. The reason that corresponds to a transition is that because of the divergence, one must treat the density of particles in the ground state separately from the above expressions for :

(51) |

where is the density of particles in the ground state and is the integral of . When , vanishes. The number of particles in the ground state doesn’t actually vanish, but rather it is no longer proportional to the volume, so that effectively vanishes.

In the pseudo-energy description (17) of the filling fraction , the condition (48) is just

(52) |

Eq. (23) then leads to the equation for : . In terms of the fugacities the critical point is the simple relation

(53) |

At the critical point, the arguments of the polylogarithms are , and if , then . (Eq. (41)). This in turn implies that the critical temperature and density are independent of the coupling :

(54) | |||||

(55) | |||||

(56) |

Though the critical temperature and density do not depend on the coupling to this order, the free energy at the critical point has corrections that do:

(57) |

The function has a simple pole at , and this has a physical significance in the present context. In particular, it implies that the critical density is infinite and the critical temperature is zero in to the order we have calculated. This means that in Bose particles have a stronger tendency to Bose-Einstein condense in comparison with . This simple pole at can thus be interpreted as a manifestation of the Mermin-Wagner theorem, which states that finite temperature continuous phase transitions are not possible in Mermin . Another way of viewing this is that in , bosonic particles behave more like fermions and don’t readily Bose-Einstein condense. This fact is ultimately responsible for why one needs to treat bosonic particles with fermionic exclusion statistics in as far as their quantum statistical mechanics is concerned. See for instance Gurarie . For relativistic models also, it appears that only the fermionic thermodynamic Bethe ansatz equations are consistentKlassen ; Mussardo .

## V Fermi gases and the Fermi energy

In this section we consider fermionic particles with the same constant kernel as in the last section. To obtain such a kernel from the second quantized hamiltonian one needs at least two species of fermions since , however for simplicity we consider only one species.

The integral we need is now:

(58) |

It is important to realize that contrary to the bosonic case, eq. (40), the above integral is valid when throughout the critical strip and is given by the above formula where:

(59) |

In physical terms, fermions are much more stable than bosons in because of the Pauli exclusion principle.

The pseudo-energy still takes the form (37), and the saddle point equation (23) now leads to the equation:

(60) |

The density and free energy can be expressed as:

(61) | |||||

In the fermionic case, the considerations of the last section are replaced by the concept of a Fermi surface. As we’ll see, mathematically there are analogies between the existence of the Fermi surface and Bose-Einstein condensation. The Fermi energy is defined as the uppermost energy that is occupied, i.e. has . In the limit of zero temperature and no interactions, is a step function: for and zero otherwise, where . At finite temperature the sharp step is broadened.

The formulas eqs. (60,61) can be used to study the effect of interactions and finite temperature on . The Fermi energy naturally can be defined as the point where , i.e. where :

(62) |

The latter implies:

(63) |

The Fermi energy may now be expressed in terms of the density :

(64) |

The above formula determines as a function of and . As the temperature goes to zero, one can use

(65) |

to obtain

(66) |

In the above formula is equivalent to well-known resultsLandau .

## Vi Bose and Fermi gases

In this section we consider the case which is rather special. Refering to eq. (44) the coupling is temperature independent in , and is in fact dimensionless. This implies that in the scattering length should simply be replaced by a dimensionless coupling we will denote as . Many of the previous formulas are especially simple due to the identity . We also describe a strong similarity to the conformally invariant limit of relativistic systems in one lower dimension.

### vi.1 Bosonic case

The equation (42) for becomes algebraic:

(67) |

The density is simply a logarithm

(68) |

and the free energy is expressed in terms of the dilogarithm:

(69) |

The density is well defined and positive at zero chemical potential , so long as . The equation (67) has a real solution in this range as long as , i.e. as long as the gas is repulsive. This observation means that the bosonic instability described in section IV can be cured by repulsive interactions. For the remainder of this section we set and denote simply as . Let us define as the following coefficient in the free energy:

(70) |

This is analagous to the relativistic case in , where one defines where in that context is the Virasoro central chargeCardy ; Affleck . In terms of the solution to the equation (67), is given in terms of Roger’s dilogarithm:

(71) |

where

(72) |

Interestingly, the above formula for in terms of is identical to the formulas that arise when one studies the conformal limit of relativistic TBA systems in ZamoTBA ; Klassen .

The function is known to satisfy the following functional relationsLewin :

(73) | |||||

Using the first relation, one sees that . This, and the free fermion and free boson cases () were known to EulerEuler . Landen found more relations as followsLanden . If is a root to the polynomial equation , then the above functional equations become linear equations for with argument and . Let us chose which is the golden mean. When is positive, the above relations imply that for special values of the coupling , is a rational number:

(74) | |||||

The above relations may be viewed as results in additive number theory by virtue of eq. (39).

Since the pressure , the coefficient is a measure of the pressure of the gas. One observes from eq. (74) that as increases the pressure decreases. This is expected on physical grounds: larger means stronger repulsive interactions so the gas is less dense.

### vi.2 Fermionic case

In this case the equation that determines (at zero chemical potential) is

(75) |

The density takes the form

(76) |

The free energy has the same form as in eq. (70) where now

(77) |

Because of the Pauli exclusion principle for fermions, the fermionic gas is more stable than the bosonic one. First note that unlike the bosonic case, the eq. (75) continues to have real solutions even if the interactions of the gas are attractive with . Another important feature is that whereas in the bosonic case has a branch cut along the real axis from 1 to , there are no branch points for along the negative axis. So for fermions, the free energy is well defined for any real .

The special rational points of the bosonic case also have a fermionic version:

(78) | |||||

Note here also that increasing decreases the pressure since the gas is either less attractive or more repulsive.

### vi.3 Many mixed particles

For far we have only considered a single bosonic or fermionic particle. It is straightforward to extend this to many types of particles of mixed statistics. Let and denote the mass and statistics of the -th particle. In the two-body approximation, we consider the following contribution to :

(79) |

where is the scattering amplitude of with types of particles. The saddle point equation now reads:

(80) |

For a constant kernel the above equation has the solution:

(81) |

In the equation satisfied by is again algebraic:

(82) |

where and are dimensionless coupling constants that parameterize the kernel . The density of the a-th species is given by

(83) |

The total free energy is given by eq. (70) with

(84) |

where is the bosonic expression (71) and the fermionic one (77).

The exists a vast number of examples where certain choices of , , and lead to rational . For instance, one can translate known fermionic relativistic systems in to the present non-relativistic context. Some of the latter are known to be related to root systems of Lie algebras. (See for instance Klassen .) There are also many more examples that follow from results in Lewin , not all of which have relativistic analog. For purposes of illustration, consider a two-particle theory with one boson and one fermion, of equal mass , and with . This structure suggests a supersymmetric theory. The are solutions of

(85) |

which implies . It can be verified that this theory has .

## Vii Quasi-periodic kernels and the Riemann hypothesis

In order get into the critical strip of with , inspection of our work so far suggests that one could try to analytically continue in the spacial dimension to complex values. However it would remain unclear what complex actually means physically. A more physical approach is to consider non-constant, quasi-periodic kernels in fixed dimension an integer. As we will show, is sufficient to cover the whole critical strip. Because of the previously discussed bosonic instabilities for , one must deal with a fermionic gas. In the next subsection we will simply hypothesize a certain kernel and work out its consequences. In the subsequent subsection we will show how to obtain such a kernel from a second quantized hamiltonian with a 2-body potential that is also quasi-periodic.

### vii.1 Quasi-periodic kernel in

We assume that the 2-body potential in eq. (25) is translationally invariant, i.e. it depends only on the difference . In momentum space this implies that the kernel also depends only on the difference:

(86) |

We also assume rotational invariance so that depends only on . Let us suppose there exists a hamiltonian which leads to the following kernel:

(87) |

where is assumed to be a complex number and is a constant. Note that the lowest order constant scattering length approximation vanishes as long as .

The saddle point equation, with zero chemical potential, reads:

(88) |

At low temperature, can be approximated near , where it continues to take the form eq. (37), with . The constant must satisfy the equation:

(89) |

where and as before . The density and free energy however have the same expressions as in section IV specialized to :

(90) | |||||

There are some trivial solutions to eq. (89) which arise when the prefactor in eq. (59) equals zero. We can remove them by chosing . We have divided by in order to remove another trivial zero which can arise from . The coupling constant then becomes:

(91) |

We can now give a clear meaning to a zero of the zeta function . Because of the relation (59), when is a zero, then a solution to eq. (89) is . This solution exists for any temperature T. By plotting the left and right hand sides of eqn. (89), one sees that in general there is another solution at , but the latter depends on , and in fact for some this solution disappears. Thus for the generic solution, this means there are no corrections to the free energy at arbitrary temperature , i.e. the density and free energy are the same as in a free (non-interacting) theory:

(92) | |||||

Note that the density is still positive since is negative: . The pressure is also positive: .

The Riemann hypothesis would then follow from the line of reasoning: If (i) The leading contributions to the pressure of the gas are the ones calculated in this work, (ii) There is a hamiltonian that leads to the -dependent quasi-periodic kernel (87), and (iii) Non-zero interactions necessarily modify the pressure of the gas over a range of temperatures, then can have no zeros. In the next subsection we address (ii).

### vii.2 Real space potentials

We now show that there are indeed real space hamiltonians in that lead to the above quasi-periodic kernel. Let be the Fourier transform of the 2-body potential:

(93) |

Using eq. (30) one finds

(94) |

We have included the statistics parameter in order to point out some features.

Consider the following potential:

(95) |

where is a constant. If is real and then the potential is quasi-periodic in :

(96) |

This model is not known to be integrable, unlike the case of a delta-function potential, thus there is no known exact thermodynamic Bethe ansatz. The lowest order corrections are the ones calculated in the last section.

When the particles are fermionic with the term in eq. (94) precisely renders the kernel well behaved as . The kernel would be singular if the particles were bosonic. The kernel is then given by the following integral:

(97) |

This integral is convergent if the following condition is satisfied:

(98) |

If the above condition is met, the kernel is the following:

(99) |

Finally we can chose in order to obtain the kernel in eq. (91). Using the identity

(100) |

this fixes to be:

(101) |

The condition (98) is precisely what one needs for the RH. What is then the meaning of the known zeros at ? These are models with the kernel (87) that give vanishing leading corrections to the pressure. The calculations of this section show that such a kernel does not arise in a convergent manner from a real space potential since the condition (98) is violated. Note that the function in eq. (99) develops a pole at , which suggests that an additional low-energy regularization could still lead to sensible models with the kernel (87) that provide physical realizations of the Riemann zeros.

## Viii Conclusions

We have shown how the formalism developed in AndreSthermo can lead to new results for the quantum statistical mechanics of interacting gases of bosons and fermions. Our main results were summarized in the introduction.

Clearly our most interesting result is the formulation of the Riemann hypothesis in the present context. We have essentially given a physical argument from which it follows. In order to develop this argument into a rigorous mathematical proof, one mainly needs to give a firm foundation to the theoretical methods used here, namely the framework developed in AndreSthermo . One also needs to understand more rigorously how the contributions considered in this paper are the leading ones, i.e. one needs clearer control of the approximations we have made.

## Ix Acknowledgments

I wish to thank Germán Sierra for many discussions on the Riemann hypothesis in connection with our work on cyclic renormalization group flows.

## X Appendix

In this appendix we explain how the duality of Riemann’s zeta function can be understood as a special modular transformation in a Lorentz-invariant theory.

Consider a free quantum field theory of massless bosonic particles in spacetime dimenions with euclidean action . The geometry of euclidean spacetime is take to be where the circumference of the circle is . We will refer to the direction at “”. Endow the flat space with a large but finite volume as follows. Let us refer to one the directions perpendicular to as with length and let the remaining directions have volume .

Let us first view the direction as compactified euclidean time, so that we are dealing with finite temperature . As a quantum statistical mechanical system, the partition function in the limit is

(102) |

where and is the free energy density. Standard results give:

(103) |

The euclidean rotational symmetry allows one to view the above system with time now along the direction. In , interchanging the role of and