## LOG#169. A two level problem.

This post is the solution of the following problem: pick an atom with two energy levels. They have a transition wavelength of 580nm. At room temperature are in the lower state.

1) How many atoms are in the upper state, if we have thermal equilibrium?

2) Suppose instead we had atoms pumped into the upper state, with atoms in the lower state. This is a non-equilibrium state. How much energy (in Joules) could be released in a single pulse of light if we restored the equilibrium?

Data: Boltzmann factor is and

Solution to 1). If is the higher energy state and is the lower energy state, then the energy difference between these two atomic states can be calculated using the transition wavelength (580nm), i.e.,

But

Now, according to the Boltzmann distribution, the population in any state is given by

Therefore, the ration of and in the thermal equilibrium should be

and thus

Using the giving value of , we get

and that is a very tiny number of atoms. This is showing to us that the energy gap between the given atomic states at room temperature is so large that almost all the electrons “choose” to stay in the lower state and hardly we will find electrons in the upper state.

Solution to 2). When we pump atoms into the upper state, we create a non-equilibrium atomic state. Energy will be released up until the equilibrium is restored (in fact, this is the working principle of the laser, in a simplified fashion!). We obtain now

and we are interested in the number of atoms which restores the thermal equilibrium. Since the ratio should be

remains the same (the gap width should not change), we find

and then

The number of atoms that contribute to restore the thermal equilibrium is

The energy released in a single monochromatic pulse would be

## LOG#168. D-dimensional laws(III).

The question of the origin of mass is one of the more important issues in theoretical physics. The existence (or not) of extra dimensions of space and time will likely affect to the final solution of this unsolved problem.

The topology of extra dimensions, and specially their “compactification” to real space and effective 4D=3d+1 world, plays a crucial role in some physics beyond the Standard Model (SM). The construction of unified models of all interactions from the purely geometric viewpoint is in the target since Einstein seminal work about the nature of the gravitational field. It also arises naturally in superstring theory, M-theory and other BSM models. In those models, particle mass stays as a problem. Nobody can deny that.

Kaluza-Klein theories (KK) are based on a general mechanism for mass creation and unification through the use of a compactification with periodic space-like dimensions (or time-like dimensions if we are general enough). The original field functions depend on all space-time coordinates, but the ordinary field functions in 4D spacetime are considered as effective result by some kind of integration (and Fourier analysis) over the extradimensional spacetime.

Suppose, for simplicity, we treat only one single extra dimension (you can extend the analysis for several extra dimensiones with suitable mathematical background). Let us introduce coordinates with , where the greek indices run over usual 4D spacetime Lorentz indices (0,1,2,3 or 1,2,3,4 if you prefer). A periodicity function condition reads

where is some parameter function depending on the scale L, the compactification length. Let us write

Define

then you can write

and

If , is real and thus and .

As we have said above, you can generalize the extra dimensional argument in a straightforward fashion. Write and , with . You get

The periodicity condition generalized to the whole extra dimensions (that would be a higher dimensional torus):

and so

with

The general KK procedure is simple. Start from a general D=4+d dimensional Lorentz invariant lagrangian and the action from the field defined as

and where

is the extradimensional world volume. Then perform over the whole extradimensional space some Fourier transformation or develop the field in terms of suitable eigenfunctions. The minimal action principle for provides the Euler-Lagrange equation

It shows that it leads to a Klein-Gordon field equation

if we define the effective field

Without treating all the possible cases, we will focus our attention on 3 cases: scalar fields, spinor fields and vector fields with extra dimensions. The most “complex” cases (since we do not know yet their quantum particles from experiments) of the Rarita-Schwinger fields (spin 3/2) and the gravitational spin 2 case (beyond higher spin extensions) will be leave for future treatment here.

1. Scalar field in extra dimensions

The free neutral scalar field can be described with a lagrangian

(1)

or equivalently

(2)

and where , and is a Minkovski metric for the extra dimensional world. It reads

We can write

(3)

It yields a Klein-Gordon (KG) equation

(4)

for some effective field

with

(5)

Remark: the squared mass

is positive if ALL the extra dimensions are space-like. It could be negative (tachyon-like!) if there exists extra time-like dimensions.

For a charged scalar field, a similar argument provides

(6)

(7)

(8)

and from here you get

2. Spinor field in extra dimensions

This case must be done with care, since there are some conditions for the existence of spinors in D-dimensional spacetime. We will neglect the subtleties related to consider the different kind of spinor fields at this moment(Dirac, Weyl, Majorana, Majorana-Weyl,…)Suppose that in (4+d)-spacetime the spinor field is described by certain component function with a free lagrangian

(9)

and where are the (4+d) Dirac (Clifford algebraic) matrices obeying the anticommutation rules (Clifford algebra):

We write

The expanded spinorial lagrangian reads

(10)

From here, we obtain

(11)

Acting to the left of this last equation with the operator

and using the Clifford algebra relations defined above, we obtain a KG like equation

and hence

Remark: if all the extra dimensions are space-like, if all the are real and we get tachyonic modes if there are one or several “big enough” extra time-like dimensions.

3. Vector field in extra dimensions

Finally, the case of a vector field. We only consider here a single extra dimension and a neutral vector field , satisfying the condition

and

The free vector field is described by the lagrangian

(12)

and thus

(13)

and where

The lagrangian expansion reads

(14)

Define a new physical vector field degree of freedom , with

It shows that the previous lagrangian can be rewritten in terms of as follows:

(15)

with . Finally, the lagrangian produces the KG field equation

tied to the effective vector field

and the squared vector field mass

Simple induction produces the mass squared vector field formula

Again, it is remarkable that the negative mass of a vector field depends on the number of extra time-like dimensions and its relative “strength”. That is, the existence of tachyonic modes (negative mass terms) is directly related to the existence of extra time-like dimensions. This is a key point that sometimes is not easily found in technical books since they neglect (usually) the (quantum stability) issues of having tachyonic modes, so they are usually ignored. But I like to talk about uncommon ideas like this one in my blog ;).

See you in my next blog post!!!!!!!

## LOG#167. D-dimensional laws(II).

Let me begin this article in D=d+1 spacetime. We are going to study quantum gases and their statistics in multidimensional space. Usual notation:

In D=d+1 spacetime, the massless free ideal relativistic gas satisfies, as we will show, certain relations between thermodinamical variables. For instance,

or

if the dispersion relationship is in d-dimensional SPACE. The Bose-Einstein integral reads

(1)

We define and the energy density

(2)

Generally, we will work with natural units (they will be reintroduced if necessary) and with zero chemical potentical and massless particles. We will explore the issue of massive particle statistics though.

Spherical coordinates (in d-dimensional euclidean space they have d-1 angles ) introduce a (d-1)dimensional solid angle

and where

and

for the solid angle . Moreover, a trivial calculation

For a dispersion relationship in d-dimensional space, we have

and where

The pressure gives us the Stefan-Boltzmann law in higher dimensions

(3)

since

We are ready to study the phenomenon of Bose-Einstein condensation (BEC) for ideal massive (relativistic and non-relativistic) bosonic gases. Take the dispersion relation to be now

and where NR denotes non-relativistic, UR denotes ultrarelativistic (massless relativistic or almost massless relativistic). Now, but the ideal bosonic gas in a box of size L, with

The critical temperature of the BEC, is approache when and . The number density will be then

(4)

Define

where with the Bose integral diverges. Then, the non-relativistic bosonic BEC temperature reads

(5)

In 3d we get the known result

where we have used .

In the case of the ultrarelativistic (massless, almost massless) case, we obtain

(6)

In the 3d case, we get the very well result

and where is the zeta value of 3.

Remark:

The 2d UR case HAS a critical temperature (unlike the 2d non-relativistic case, where BEC does NOT exist) . Indeed, you can easily check that

where .

Remark (II):

BEC does depend not only on the number of dimensions but also on the density of states, i.e., it is highly dependent on the dispersion relationship we use!

An additional important issue is the following. If we allow , then pairs boson-antiboson can be created. In particular, we have that

for each particle. Moreover, you have

with + for bosons and – for antibosons, so

and then

with . Thus, we get

Case 1. Low T, with . Then , as we would expect.

Case 2. High T, with . Then, the critical temperature DOES change to take into account the boson-antiboson pair creation. It yields

(7)

Note that the UR boson-antiboson massive case is not equal to the UR boson (massless) case, even in 3d! Indeed, you find that

In fact, with no antiboson (massless), the case provides the following thermodynamical quantities:

When antibosons are present, these integrals become nastier and more complicated:

Challenge for eager readers:

Take the density of states

and take the UR limit to get and prove

and calculate/check the critical temperature

obtaining the value of the “constant” above.

By the other hand, we can also study the fermionic gas in D-dimensional spacetime, d-dimensional space. In order to simplify the discussion, we are going to study only the non-relativistic (NR) ideal gas. I will study the relativistic Fermi gas in a future post because it is important in extremely degenerate systems, as some particular kind of stars. The ideal non-relativistic Fermi gas in d-dimensional space has the following interesting features:

Moreover, you have

and the Fermi energy reads ():

The dimensional Fermi weights are, for D=d+1 dimensional spacetime and the NR and UR case

The non-relativistic (NR) fermionic (or fermi) gas has a dispersion relationship

and the energy in terms of Fermi quantities reads

and the density of states

Thus, the energy density will be

with the thermal wavelength

Furthermore

where is the fugacity. The Fermi function reads

and it uses the polylogarithm as well!!! Wonderful, isn’t it? The average energy per fermion in d-space is

You get if , if and you also have

with

and the number density

The massless spinless bosonic particles in D=d+1 dimensions have a free energy

The Casimir energy of such a bosonic field requires

and the regularized energy in vacuum has to be

and it shows that the Riemann zeta functional equation holds iff

This striking consequence and relationship between the vacuum structure and pure mathematics is fascinating and not yet completely understood. But this will be a topic for a future discussion here.

See you in my next blog post!

## LOG#166. D-dimensional laws(I).

My next thread of nested articles are aimed to explore (shallowly, only) the issue of extra dimensions (of space and time!).

Let me begin with the Gauss law for the electric field. The electrical flux in D-dimensional space reads, for radial symmetry,

up to a multiplicative constant, related to in D=4 spacetime. There, is the radial electric field, is certain gaussian hypersurface containing the electric charge Q. If we take a hypersphere, so equals the surface of the (d-1)-sphere, we would obtain a Coulomb-like law in D=d-1 dimensions:

(1)

Note that we have used units in which for simplicity. We recover the 3D-space, 4D-spacetime result using the above expression with care. You can also check that the electric field can be derived from the potential through the usual definition

How can we modify this law if we include the possibility of having massive gauge bosons and their interactions? If there is a massive gauge boson exchange, then the short-range Yukawa interaction provides the following potential in (d-1)-dimensional space:

(2)

and where is the Bessel function of 2nd type.

Let me remember you some facts about hyperspheres. The (d-1)D hypersphere surface can be computed as follows

and thus

The volume for the hypersphere is

where the (d-1)-spherical shell volume is constrained by the relationship

and the solid sphere is instead the constrain

In general (d+1)=D spacetime, for a single point particle, we write the D-dimensional Coulomb-like electric field as

(3)

By dimensional analysis, we get that

Moreover, you have

to be more precise for any .

Thus, the Coulomb law between two point particles in extra dimensions (XD), D=(d+1)D spacetime reads

(4)

and thus the electrostatic (d+1)D field and scalar potential are given by

(5)

(6)

Magnetism can be also generalized to higher dimensions. A more appropiate language would be the language of differential forms (exterior calculus) and/or geometric calculus (geometric algebra) in order to include not only point particles, but also defects such as branes and their electric/magnetic charges, the notion of duality and more (such as topological terms like Chern-Simons and their nontrivial couplings). But I will write about it in the future, I promise, …Today we will focus on a very basic high-school/lower undergraduate calculus…Forgive me this today. I think it has its advantages for pedagogical reasons to begin at basic level…

The (d+1)D magnetic field law can be written as follows

(7)

A dimensional check provides that

and the above law can be understood as the (d+1)D magnetic field inducing a magnetic force. You can also check that

For one steady current over other parallel (both filamentary) the magnetic force per unit length equals to

(8)

Some simple tensor magnetic field equations in (d+1)D, with current distribute over a one dimensional, a two dimensional or a three dimensional cross section, can be guessed. For a 1-dimensional cross section

(9)

For a two dimensional cross section current

(10)

and for the three dimensional cross section current

(11)

and where are current densities, and is along the current .

Remark: In the 3+1 world we recover the classical and common result and

as you can easily check out. For a XD (eXtra Dimensional) world you have, in general, certain lagrangian living on D=d+1 spacetime. A (d+1) Poisson equation reads

Free space solutions of the Laplace equation above requires a Fourier transform, so we calculate

and it implies an integral representation

and where is the (d-1)D angular element ((d-1)-dimensional solid angle). We can define spherical coordinates

so we have

and

Performing the integration carefully, taking into account the surface of the hypersphere, we have an integral

and from this, we recover the classical (d+1)-dimensional potential given above, i.e.,

Q.E.D.

## LOG#165. Rogers-Ramanujan identities.

There are some cool identities, very well known to mathematicians and some theoretical physicists or chemists, related with Ramanujan. They are commonly referred as Rogers-Ramanujan identities (Rogers, 1894; Ramanujan 1913,1917 and Rogers and Ramanujan, 1919). They are related to some objects called basic hypergeometric functions, some q-analogues of the classical hypergeometric functions that some people study in calculus and the theory of special functions. We write them as follows:

(1)

(2)

and where . It shows that and are modular functions of . These identities arise in some places of superstring theory, and in the hard hexagon model of statistical mechanics. Moreover, they are the key ingredients to derive the Ramanujan’s continued fraction

(3)

that were some of the most remarkable findings that Ramanujan sent to the English mathematician Hardy in the United Kingdom at the first of the 20th century…

May Ramanujan be with you!

## LOG#164. Theta functions.

Hi, there! We are going to explore more mathematical objects in this post. Today, the objects to study are theta functions. A prototype is the Jacobi theta function:

(1)

where and . It satisfies the functional equation

(2)

and

(3)

. The Jacobi theta function is related to the Riemann zeta function. With

we have

(4)

We can also define some auxiliary Jacobi theta functions that we found in the classical literature about this complex subject

where is sometimes called the nome. Auxiliary theta functions can be written in terms of the nome, and if

they satisfy the following Jacobi identities

The Jacobi theta function is the fundamental solution to the 1D heat equation with spatially periodic boundary conditions. That is, if we write:

we find that it verifies

with

So, the Jacobi theta function becomes the Dirac comb when t approaches to zero. Some very interesting values of the Jacobi theta function are related to the gamma function:

The notion of theta function can be generalized. Let F be a quadratic form in n-variables. Then, we define

It is a modular form of weight n/2. The Ramanujan theta function arises in a similar way as well:

(5)

and where is a n-dimensional complex vector, T denotes the transpose, , such as . Note that the n-dimensional analogue of the modular group is the sympletic group. And the double cover of the symplectic group is the metaplectic group. It can be defined over either real or p-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles.

The metaplectic group has a particularly significant infinite-dimensional linear representation, the Weil representation. It was used by André Weil to give a representation-theoretic interpretation of theta functions, and is important in the theory of modular forms of half-integral weight and the theta correspondence.

The fundamental group of the symplectic Lie group is infinite cyclic, so it has a unique connected double cover, which is denoted  and it denotes the metaplectic group.

The metaplectic group is not a matrix group: it has no faithful finite-dimensional representations. Therefore, the question of its explicit realization is nontrivial. For more details, and the completion of all this material, you can read the whole article about the metaplectic group in the wikipedia.

The Ramanujan theta function satisfies a beautiful functional equation

(6)

A much more sophisticated (and modern) concept of theta function is that of mock theta function and mock modular forms.But it will be the topic of a future post…Some day.

See you in my next blog post!!!!!

## LOG#163. Q-stuff and wonderful functions.

In this blog post I am going to define and talk about some interesting objects. They are commonly referred as q-objects in general.

The q-Pochhammer symbol is the next product:

(1)

with . The infinite product extension is also very popular

(2)

and it is analytic in the unit disc, with being the Euler’s function, important object in combinatorics, number theory and the theory of modular forms.

The q-Pochhammer symbol satisfies a big number of identities. I like mostly four of them:

(3)

(4)

(5)

And the fourth is the so-called q-binomial theorem

(6)

Interpretation: the coefficient of in the expansion

is the number of partitions of m into at most n parts. Moreover, if we write

that is the number of partitions of m into n or n-1 parts when we read off the coefficient in . The q-Pochhammer function admits multiple arguments in the following way:

The q-Pochhammer symbol can be related to other q-objects. We define first the q-numbers and the q-factorial. The q-numbers are defined as

and the q-factorial is

Now, we can even define a q-deformed version of the traditional derivative. It is called the q-derivative or Jackson’s derivative:

(7)

It satifies some conventional rules and some deformed variants of the classical derivative

Moreover, we also have

The Taylor expansion analogue also exists:

There are also some theta functions to explore. The q-theta function is

(8)

and with this yields

(9)

The Ramanujan theta function is a fascinating object I wish to show you:

(10)

with . It appears (and can be used in some applications of) in critical bosonic string theory, superstring theory and M-theory. This Ramanujan theta function satisfies a beautiful identity called Jacobi triple product

Some additional identities of this Ramanujan theta function are:

(11)

(12)

and

(13)

so the Euler function is a particular case of the Ramanujan theta function! They are also related to the Dedekind eta function. The Jacobi theta function may be written in terms of the Ramanujan theta function as well:

By the other hand, the Jacobi triple product identity is generally something more “general”. It is the mathematical identity:

and it is defined for arbitrary complex numbers such as and . The two more elegant forms of this Jacobi triple product identity are bound to the Ramanujan theta function we have defined above or in terms of the q-Pochhammer symbols

where is the infinite q-Pochhammer symbol and

in terms of Ramanujan theta function, as we have already seen previously. Aren’t you amazed by those formulae? You should! They are strikingly appealing and beautiful.

May the q-functions be with you!!!!

## LOG#162. Polylogia flashes(IV).

In this final post (by the moment) in the polylogia series we will write some additional formulae for polylogs and associated series.

Firstly, we have

(1)

and now, if

(2)

(3)

The next identity also holds

The Bose-Einstein integral can be rewritten for as follows

and

The polylog has the following asymptotic series. If we have

(4)

(5)

Euler’s dilogarithm or Spence’s function has nice features. Interestingly, computer algebra systems generally define the dilogarithm as , thus be aware with the definitions you read, use and write. In our case:

(6)

Moreover, for you can find

(7)

(8)

and we have the Abel identity:

(9)

Euler’s reflection formula with follows up

(10)

The pentagon identity, with and is a mutation of the Abel identity

Landen’s identity is also beautiful. It arises if we write in the Abel identity and we square the relationship we get:

with . We have an inversion formula for this too

with , or

with .

The mathematician Don Zagier has stated the following sentence “(…)The dilogarithm is the only mathematical functionwith a sense of humor(…)”. Have a look at the following values of Euler’s dilogarithm:

Moreover, suppose that is the negative golden mean ratio, then

If we also have

If , we have

and

There are also polylog ladders much more complex than the above identities. Let us define

Then, a wonderful result by Coxeter (1935) is the next identity

(11)

and Landen also derived

(12)

Now, we can write some multiplication theorems. The duplication identity

(13)

Gauss wrote the next sum, a discrete Fourier transform

(14)

The Kummer’s identity for duplication reads

(15)

Moreover, if , we can derive the result

and where is the Bessel function with .

Periodic zeta functions are defined by

(16)

and thus

Finally, we end this series with some Hurwitz zeta function series and identities

See you in my next blog post!

## LOG#161. Polylogia flashes(III).

In the third post of this series I will write more fantastic identities related to our friends, the polylogs!

(1)

and by analytic continuation that equation can be extended to all . In fact

(2)

such as , since we define

(3)

and the equations

(4)

(5)

(6)

Duplication formula for the polylogarithm:

(7)

Connection with the Kummer’s function can be established

(8)

(9)

and thus

(10)

We also have

(11)

Some extra values of (negative) integer polylogs that are rational functions or logarithms

(12)

(13)

(14)

(15)

(16)

(17)

And more generally, we have the general formulae

(18)

(19)

and where are the Stirling numbers of the second kind.

(20)

Furthermore,

(21)

and where are the eulerian numbers.

We write now some interesting values of the polylog you will love too

and where

with .

The polylog and other functions can be also be related, as we have seen:

(22)

(23)

(24)

where is the Dirichlet beta function. The complete Fermi-Dirac integral is also polylogarithmic

The incomplete polylog is also interesting

and the incomplete gamma function is defined by

(25)

with , , . Moreover,

(26)

There is also a formula called inversion formula

(27)

and for

(28)

The expression

implies that

(29)

with

if and if . The following feynmanity (nullity) identity holds as well

(30)

and

(31)

for and

(32)

for . Polylogs and Clausen functions are related (we already saw this before)

(33)

The inverse tangent integral is related to polylogs

(34)

(35)

(36)

(37)

The Legendre chi function is also related to polylogs

(38)

The incomplete zeta function or Debye functions are polylogs as well

(39)

(40)

(41)

Now, we will write some polylog integrals

(42)

This last integral converges if and . It is the Bose-Einstein distribution!

The Fermi-Dirac integrals read

(43)

For and excepting and we have

and

(44)

with residue equal to

See you in the next polylog post!

## LOG#160. Polylogia flashes(II).

The polylogarithm or Jonquière’s function is generally defined as

Do not confuse with the logarithm integral in number theory, which is

such as

and

. In fact, notation can be confusing sometimes since the european is sometimes written as and

The polylogarithm (or polylog, for short) is sometimes written as , i.e.,

so…Be aware with notations and the meaning of the symbols! The polylog is related with the Lerch trascendent in the following way

(1)

The polylog is a wonderful function. It is ubiquitous in Physics, Chemistry and Mathematics…For instance, it arises in:

1st. Feynman diagram integrals, renormalization, and, in particular, in the calculation of the QED (Quantum Electrodynamics) corrections to the electron gyromagnetic ratio, supegravity amplitudes and other quantum scattering problems.

2nd. Quantum statistics. The Fermi-Dirac and the Bose-Einstein statistics can be written in terms of polylogarithms.

3rd. Vacuum effects in strong fields and quantum gravity. Non-perturbative effects in QFT (Schwinger effect, instanton effects and others) can be handled with these incredible functions.

Let us analyze the case of quantum statistics a little. The Fermi-Dirac statistics/distribution can be written as follows

(2)

The Bose-Einstein statistics/distribution can be written as follows

(3)

The polylog can  be easily related to the Riemann zeta function

In fact, “colored” polylogs do exist. We will write on them later…

Remark: does appear in the 3rd order correction to the gyromagnetic ratio of the electron in QED.

Polylogs have some really cool and interesting properties and values. Firstly, the polylog is itself a polylog when it is derived (so it has a striking similarity with the classical exponential function):

(4)

This self-similarity is certainly suggestive for certain equations of mathematical physics. Moreover, some surprising identities of the polylogarith are like this one (Bayley et al. proved it):

(5)

No algorithm is known yet for integration of polylogarithms of functions in closed form. By the other hand, polylogs can be also defined for negative values of “s” and integer numbers, i.e., for all

In fact, we have

and where

is an eulerian number.

Polylogs also arise in the theory of generalized harmonic numbers

with . There are some special polylogs with special (perhaps subjective) beauty

Note that these are rational functions! We also have

Some interesting known values of the polylog

No higher formulas are known for if

Euler’s dilogarithm is defined to be

It satisfies many functional identities and the dilogarithm values are interesting theirselves in many calculations. The trilogarithm or trilog can also be written to be

and where

and the beautiful result

(6)

and thus we obtain another wonderful result

The multidimensional polylog is defined as

(7)

Furthermore, the colored polylog is

(8)

The Nielsen generalized polylogarithm is

such as

and it is computed as Polylog[n,p,z] sometimes. The Nielsen-Ramanujan constanst are also beautiful:

and

with

, ,…

Finally, we are going to define higher order prime zeta functions. Recall that the classical Riemann zeta function and the prime zeta function have already being defined as

and thus, the k-th order prime zeta function should be written as

Moreover, the prime zeta function of infinite order should be . The iterated -power of the Riemann zeta function is

We could call it the omega zeta function.

See you in my next polylog post!!!!!!