LOG#169. A two level problem.

lasersword Laser_beam

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 5\cdot 10^{20}atoms are in the lower state.

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

2) Suppose instead we had 9\cdot 10^{20} atoms pumped into the upper state, with 5\cdot 10^{20} 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 \exp\left(-E/k_BT \right) and k_B=1\mbox{.}38\cdot 10^{-23}J/K

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

    \[E_b-E_a=hf=\dfrac{hc}{\lambda}=\Delta E\]

    \[\Delta E=3\mbox{.}43\cdot 10^{-19}J\]

But

    \[k_BT=1\mbox{.}38\cdot 10^{-23}J/K\cdot 300K=4\mbox{.}10^{-21}J\]

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

    \[N_i=\exp\left(E_i/k_BT\right)\]

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

    \[\dfrac{N_b(eq)}{N_a(eq)}=e^{-\frac{\Delta E}{k_BT}}=1\mbox{.}1\cdot 10^{-36}\]

and thus

    \[N_b(eq)=N_a(eq)\cdot 1\mbox{.}1\cdot 10^{-36}\]

Using the giving value of N_a, we get

    \[N_b(eq)=5\mbox{.}5\cdot 10^{-16}\]

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

    \[N_a(non-eq)=5\cdot 10^{20}\]

    \[N_b(non-eq)=9\cdot 10^{20}\]

    \[N(non-eq)=N_a(non-eq)+N_b(non-eq)=14\cdot 10^{20}\]

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

    \[\dfrac{N_a}{N_b}=1\mbox{.}1\cdot 10^{-36}\]

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

    \[N_a(non-eq)+N_b(non-eq)=14\cdot 10^{20}\]

    \[N_b(non_eq)(1+9\mbox{.}09\cdot 10^{35})=9\cdot 10^{20}\]

and then

    \[N_b(non-eq)=1\mbox{.}54\cdot 10^{-15}\]

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

    \[\Delta N=N(non-eq)-N_b(non-eq)\approx 9\cdot 10^{20}\]

The energy released in a single monochromatic pulse would be

    \[E=\Delta N\cdot \dfrac{hc}{\lambda}=9\cdot 10^{20}\cdot 3\mbox{.}43\cdot 10^{-19}\approx 309J\]

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

flatland

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 X^M with M=\mu,5, 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

    \[\boxed{F(X^\mu,X^5+L)=f_L(L)F(X^\mu,X^5)}\]

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

    \[\dfrac{\partial}{\partial x^5}F(X^M)=\partial_5F(X^M)=g_L(L)F(X^M)\]

Define

    \[f_L(L)=e^{Lg_M(L)}\]

    \[g_L(L)=\dfrac{1}{L}\left[\ln f_L(L)+2\pi i n\right](n\in Z)\]

then you can write

    \[f_L(L)=\rho_F(L)e^{i\theta_F(L)}\]

and

    \[g_L(L)=\dfrac{1}{L}\left[\ln\rho_F(L)+i(\theta_F(L)+2\pi n)\right]\]

If F=F^+, f_L(L) is real and thus \theta_F=0 and n=0.

As we have said above, you can generalize the extra dimensional argument in a straightforward fashion. Write X^5,X^6,\ldots,X^{4+d} and Y^a\equiv X^{4+a}, with a=1,2,\ldots,d. You get

    \[F(X^M)=F(X^\mu,Y^a)\equiv F(X,Y)\]

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

    \[F(X,Y^a+L^a)=f_F^{(a)}(L_a)F(X,Y)\]

and so

    \[\dfrac{\partial}{\partial Y^{a}}F(X,Y)=\partial_{Y^a}F(X,Y)\equiv \partial_aF(X,Y)=g_F^{(a)}(L^a)F(X,Y)\]

with

    \[f_F^{(a)}(L^a)=f_F^{(a)}(L^a)e^{i\theta_f^{a}(L^a)}\]

    \[g_F^{(a)}(L^a)=\dfrac{1}{L^a}\left[\ln f_F^{(a)}(L^a)+i(\theta_F^{(a)}(L^a)+2\pi n)\right]\]

 The general KK procedure is simple. Start from a general D=4+d dimensional Lorentz invariant lagrangian L(X,Y) and the action from the field F(X,Y) defined as

    \[S=\int S(Y)(dY)\]

    \[S(Y)=\int d^4xL(X,Y)\]

and where

    \[(dY)=dY^1\cdots dY^d\]

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 S(Y) provides the Euler-Lagrange equation

    \[\dfrac{\partial L(X,Y)}{\partial F(X,Y)}-\partial_\mu\dfrac{\partial L(X,Y)}{\partial(\partial_\mu F(X,Y))}=0\]

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

    \[(\square^2+m_F^2)F(X)=0\]

if we define the effective field

    \[F(X)=\int (dY)F(X,Y)\]

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 \Phi(X,Y) can be described with a lagrangian

(1)   \begin{equation*} L(X,Y)=\dfrac{1}{2}\partial^M\Phi(X,Y)\partial_M\Phi(X,Y) \end{equation*}

or equivalently

(2)   \begin{equation*} L(X,Y)\equiv \dfrac{1}{2}\{\partial^\mu\Phi(X,Y)\partial_\mu\Phi(X,Y)+\sum_{a=1}^d \delta_{aa}\partial_a\Phi(X,Y)\partial_a\Phi(X,Y)\} \end{equation*}

and where \partial_a=\dfrac{\partial}{Y^a}, and \delta_{ab} is a Minkovski metric for the extra dimensional world. It reads

    \[\delta_{ab}=\begin{cases}0,\mbox{if}\;\;\; a\neq b\\ +1,\mbox{if},\;\;\; a=b(timelike)\\ -1,\mbox{if}\;\;\; a=b(spacelike)\end{cases}\]

We can write

(3)   \begin{equation*} L(X,Y)=\dfrac{1}{2}\left[\partial^\mu\Phi(X,Y)\partial_\mu\Phi(X,Y)+\sum_{a=1}^d\delta_{aa}(g^{(a)}(L^a))^2\Phi^2(X,Y)\right] \end{equation*}

It yields a Klein-Gordon (KG) equation

(4)   \begin{equation*} (\square^2+m_\Phi^2)\Phi(X)=0 \end{equation*}

for some effective field

    \[\Phi(X)=\int (dY)\Phi(X,Y)\]

with

(5)   \begin{equation*} m_\Phi^2=-\sum_{a}\delta_{aa}(g^{(a)}(L^a))^2 \end{equation*}

Remark: the squared mass

    \[\boxed{m_\Phi^2=-\sum_{a}\delta_{aa}(g^{(a)}(L^a))^2}\]

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)   \begin{equation*} L(X,Y)=\dfrac{1}{2}\partial^M\Phi^+(X,Y)\partial_M\Phi(X,Y) \end{equation*}

(7)   \begin{equation*} L(X,Y)\equiv \dfrac{1}{2}\{\partial^\mu\Phi(X,Y)\partial_\mu\Phi(X,Y)+\sum_{a=1}^d \delta_{aa}\partial_a\Phi^+(X,Y)\partial_a\Phi(X,Y)\} \end{equation*}

(8)   \begin{equation*} L(X,Y)=\dfrac{1}{2}\left[\partial^\mu\Phi^+(X,Y)\partial_\mu\Phi(X,Y)+\sum_{a=1}^d\delta_{aa}\vert g^{(a)}(L^a)\vert^2\Phi^+(X,Y)\Phi (X,Y)\right] \end{equation*}

and from here you get

    \[\boxed{m_\Phi^2=-\sum_{a}\delta_{aa}\vert g^{(a)}(L^a)\vert^2}\]

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 2^{(4+d)/2} component function \Psi_\alpha(X,Y) with a free lagrangian

(9)   \begin{equation*} L(X,Y)=\dfrac{i}{2}\overline{\Psi(X,Y)}\Gamma^M\overleftrightarrow{\partial_M}\Psi(X,Y)=\dfrac{i}{2}\left(\overline \Psi\Gamma^\mu\overleftrightarrow{\partial_\mu}\Psi+\sum_{a=1}^d\overline\Psi\Gamma^{a+4}\overleftrightarrow{\partial_a}\Psi\right) \end{equation*}

and where \Gamma^M are the (4+d) Dirac (Clifford algebraic) 2^{(4+d)/2}\times 2^{(4+d)/2} matrices obeying the anticommutation rules (Clifford algebra):

    \[\{\Gamma^\mu,\Gamma^\nu\}=2\delta^{\mu\nu}\]

    \[\{\Gamma^M,\Gamma^{4+d}\}=0\]

    \[\{\Gamma^{4+a},\Gamma^{4+b}\}=2\delta^{ab}\]

    \[\overline{\Psi}=\Psi+\Gamma^0\]

We write

    \[\partial_a\Psi(X,Y)=g_\Psi^{(a)}L_a\Psi(X,Y)\]

    \[\partial_a\overline{\Psi(X,Y)}=g_\Psi^{(a)}L_a\overline{\Psi(X,Y)}\]

The expanded spinorial lagrangian reads

(10)   \begin{equation*} L(X,Y)=\dfrac{i}{2}\overline{\Phi(X,Y)}\Gamma^\mu\overleftrightarrow{\partial_\mu}\Psi(X,Y)-\mathcal{I}(g_\Psi^{(a)}(L^a))\overline{\Psi}\Gamma^{4+a}\Psi \end{equation*}

From here, we obtain

(11)   \begin{equation*} \left(i\Gamma^\mu\partial_\mu-\sum_{a=1}^d\mathcal{I}(g_\Psi^{(a)}(L^a))\Gamma^{4+a}\right)\Psi(X,Y)=0 \end{equation*}

Acting to the left of this last equation with the operator

    \[i\Gamma^\nu\partial_\nu-\sum_{b=1}^d\mathcal{I}(g_\Psi^{(b)}(L^b))\Gamma^{4+b}\]

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

    \[\left[\square^2-\sum_{a=1}^d\delta_{aa}(\mathcal{I}(g_\Psi^{(a)}(L^a)))^2\right]\Psi(X,Y)=0\]

and hence

    \[\boxed{m_\Psi^2=-\sum_{a}\delta_{aa}(\mathcal{I}(g_\Psi^{(a)}(L^a))^2}\]

Remark: m_\Psi^2>0 if all the extra dimensions are space-like, m^2_\Psi=0 if all the g_\Psi^{(a)} are real and we get tachyonic modes m_\Psi^2<0 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 V_M(X,Y), satisfying the condition

    \[V_M(X,Y+L)=f_V(L)V_M(X,Y)\]

and

    \[\dfrac{\partial}{\partial Y}V_M(X,Y)=g_V(L)V_M(X,Y)\]

    \[f_V(L)=e^Lg_V(L)\]

The free vector field V_M(X,Y) is described by the lagrangian

(12)   \begin{equation*} L(X,Y)=-\dfrac{1}{4}F_{MN}F^{MN}=-\dfrac{1}{4}\left(F^{\mu\nu}F_{\mu\nu}+2F_{\mu 5}F^{\mu 5}\right) \end{equation*}

and thus

(13)   \begin{equation*} L(X,Y)=-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}-\dfrac{1}{2}\delta_{55}\left(\partial_\mu V_5\partial^\mu V_5+\partial_5V_\mu\partial_5V^\mu-2\partial_\mu V_5\partial_5 V^\mu\right) \end{equation*}

and where

    \[F_{\mu\nu}=\partial_\mu V_\mu-\partial_\nu V_\mu\]

    \[F_{\mu 5}=\partial_\mu V_5-\partial_5V_\mu\]

The lagrangian expansion reads

(14)   \begin{equation*} L(X,Y)=-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}-\dfrac{1}{2}\delta_{55}\left(\partial_\mu V_5\partial^\mu V_5+g_V^2(L)V_\mu V^\mu-2g_V(L)\partial_\mu V_5V^\mu\right) \end{equation*}

Define a new physical vector field degree of freedom W^\mu, with

    \[W^\mu=V_\mu-\dfrac{1}{g_V(L)}\partial_\mu V_5\]

It shows that the previous lagrangian can be rewritten in terms of W^\mu as follows:

(15)   \begin{equation*} L(X,Y)=-\dfrac{1}{4}G_{\mu \nu}G^{\mu\nu}-\dfrac{1}{2}\delta_{55}g_V^2(L)W^\mu W_\mu \end{equation*}

with G_{\mu\nu})=\partial_\mu W_\nu-\partial_\nu W_\mu. Finally, the lagrangian produces the KG field equation

    \[(\square-\delta_{55}g_V(L)^2)W^\mu=0\]

tied to the effective vector field

    \[W_\mu (X)=\int_0^L dYW_\mu (X,Y)\]

and the squared vector field mass

    \[\boxed{m_W^2=-\delta_{55}g_V^2(L)}\]

Simple induction produces the mass squared vector field formula

    \[\boxed{m_W^2=-\sum_{a=1}^d\delta_{aa}g_V^2(L^a)}\]

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).

be BECtraps boseBEC PhaseTrans

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:

    \[\sum_{\vec{k}}\rightarrow V_d\int\dfrac{d^dk}{(2\pi)^d}=V_d\int\dfrac{d^dk}{(2\pi \hbar)^d}\]

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

    \[P=\dfrac{\varepsilon}{d}=\dfrac{u}{d}\]

or

    \[P=\dfrac{s\varepsilon}{d}\]

if the dispersion relationship is \varepsilon=ap^s in d-dimensional SPACE. The Bose-Einstein integral reads

(1)   \begin{equation*} I_{BE}=\int_{0}^\infty\dfrac{x^n}{e^x-1}=\Gamma(n+1)\zeta(n+1) \end{equation*}

We define n=\dfrac{N}{V_d} and the energy density

(2)   \begin{equation*} \varepsilon=\rho=\dfrac{E}{N}=\int\dfrac{d^dk}{(2\pi)^d}\dfrac{E}{e^{\beta E}-1} \end{equation*}

Generally, we will work with natural units \hbar=k_B=1 (they will be reintroduced if necessary) and with zero chemical potentical \mu=0 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 (\varphi,\theta_1, \theta_2,\ldots,\theta_{d-2})) introduce a (d-1)dimensional solid angle

    \[d\Omega_{d-1}=2\pi\prod_{n=1}^{d-2}\sin^n\theta_nd\theta_n\]

and where

    \[\Omega_{d-1}=\dfrac{2\pi^{d/2}}{\Gamma\left(\dfrac{d}{2}\right)}=\dfrac{2\left(\Gamma\left(\dfrac{1}{2}\right)\right)^d}{\Gamma\left(\dfrac{d}{2}\right)}\]

 and

    \[\int_0^\pi d\theta\sin^n\theta=\sqrt{\pi}\dfrac{\Gamma\left(\dfrac{n+1}{2}\right)}{\Gamma\left(\dfrac{n+2}{2}\right)}\]

    \[\langle\cos^2\theta_{d-2}\rangle=1-\langle\sin^2\theta_{d-2}\rangle\]

for the solid angle \Omega_{d-1}. Moreover, a trivial calculation

    \[\langle \sin^2\theta_{d-2}\rangle=\dfrac{\Gamma\left(\dfrac{d}{2}\right)\Gamma\left(\dfrac{d+1}{2}\right)}{\Gamma\left(\dfrac{d-1}{2}\right)\Gamma\left(\dfrac{d+2}{2}\right)}=\dfrac{d-1}{d}=\dfrac{D-2}{D-1}\]

    \[\langle\cos^2\theta_{d-2}\rangle=1-\langle\sin^2\theta_{d-2}\rangle=1-\dfrac{d-1}{d}=\dfrac{1}{d}=\dfrac{1}{D-1}\]

For a dispersion relationship \varepsilon=ap in d-dimensional space, we have

    \[\boxed{\varepsilon=dP(T)=\dfrac{2\Gamma\left(d+1\right)\zeta(d+1)\left(k_BT\right)^{d+1}}{\Gamma\left(\dfrac{d}{2}\right)\left(4\pi\right)^{d/2}}=\Omega_{d-1}\dfrac{1}{(2\pi)^d}\Gamma\left(D\right)\zeta\left(D\right)\left(k_BT\right)^D}\]

and where

    \[\Omega_{d-1}=\dfrac{2\pi^{d/2}}{\Gamma\left(\dfrac{d}{2}\right)}\]

The pressure P(T) gives us the Stefan-Boltzmann law in higher dimensions

(3)   \begin{equation*} \boxed{P(T)=\dfrac{\varepsilon}{d}=\dfrac{2}{d}\dfrac{\Gamma\left(d+1\right)\zeta\left(d+1\right)\left(k_BT\right)^{d+1}}{\Gamma\left(\dfrac{d}{2}\right)\left(4\pi\right)^{d/2}}} \end{equation*}

since

    \[\dfrac{\Gamma\left(d+1\right)}{\dfrac{d}{2}\Gamma\left(\dfrac{d}{2}\right)\left(4\pi\right)^{d/2}}=\dfrac{\Gamma\left(\dfrac{D}{2}\right)}{\pi^{D/2}}=\dfrac{\Gamma\left(\dfrac{d+1}{2}\right)}{\pi^{(d+1)/2}}\]

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

    \[E_k=\sqrt{c^2\hbar^2k^2+m^2c^4}=\begin{cases}mc^2+\dfrac{\hbar^2k^2}{2m}+\mathcal{O}(k^4),NR(c\hbar k<<mc^2)\\ \hbar ck\left[1+\dfrac{1}{2}\left(\dfrac{mc}{\hbar k}\right)^2+\mathcal{O}(k^{-4})\right],UR(\hbar ck>>mc^2)\end{cases}\]

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

    \[\sum_{k}\rightarrow\left(\dfrac{L}{2\pi}\right)\int d^dk=V_d\int d^dk\]

The critical temperature of the BEC, T=T_c is approache when \mu(T_c)=mc^2 and \beta_c=1/k_BT_c. The number density will be then

(4)   \begin{equation*} n=\dfrac{N}{L^d}=\dfrac{N}{V_d}=\dfrac{1}{(2\pi)^d}\int d^dk\dfrac{1}{e^{\beta_c(\vert E_k\vert-mc^2)}-1} \end{equation*}

Define

    \[d^dk=\dfrac{2\pi^{d/2}}{\Gamma\left(\dfrac{d}{2}\right)}k^{d-1}dk\]

    \[z_c=e^{\mu/k_BT_c}\]

where with z\leq 1 the Bose integral diverges. Then, the non-relativistic bosonic BEC temperature reads

(5)   \begin{equation*} \boxed{k_BT_c(NR,B)\equiv \dfrac{2\pi \hbar^2}{m}\left[\dfrac{n}{\zeta\left(\dfrac{d}{2}\right)}\right]^{2/d}} \end{equation*}

In 3d we get the known result

    \[k_BT_c(3d,NR,B)\simeq 3\mbox{.} 31\hbar^2\dfrac{n^{2/3}}{m}\]

where we have used \zeta(3/2)\simeq 2\mbox{.} 612.

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

(6)   \begin{equation*} \boxed{k_BT_c(UR,B)=\left[\dfrac{\hbar^dc^d2^{d-1}\pi^{d/2}\Gamma\left(\dfrac{d}{2}\right)}{\Gamma(d)\zeta(d)}\right]^{1/d}n^{1/d}} \end{equation*}

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

    \[k_BT_c(3d,UR,B)=\hbar c\pi^{2/3}\left(\dfrac{n}{\zeta (3)}\right)^{1/3}\simeq 2\mbox{.}017\hbar c n^{1/3}\]

and where \zeta(3)=1\mbox{.}20206 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) T_c\neq 0. Indeed, you can easily check that

    \[k_BT_c(2d,UR,B)=\hbar c\left(\dfrac{2\pi n}{\zeta (2)}\right)^{1/2}\simeq 1\mbox{.}954\hbar c n^{1/2}\]

where \zeta (2)=\pi^2/6.

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 \varepsilon=\varepsilon(p) we use!

An additional important issue is the following. If we allow k_BT>>mc^2, then pairs boson-antiboson B\overline{B} can be created. In particular, we have that

    \[E_k^2=\hbar^2c^2k^2+m^2c^4\]

for each particle. Moreover, you have

    \[E_k=\pm E_p\]

with + for bosons and – for antibosons, so

    \[(N_k-\overline{N}_k)V_d=\sum_kn_k-\overline{n}_k\]

and then

    \[N_k-\overline{N}_k=\sum_k\left(\dfrac{1}{e^{\beta(E_k-\mu)}-1}-\dfrac{1}{e^{\beta(E_k+\mu)}-1}\right)\]

with -mc^2<\mu<mc^2. Thus, we get

    \[n=\dfrac{N-\overline{N}}{V_d}=\dfrac{N-\overline{N}}{L_d}=\dfrac{2\pi^{d/2}}{\Gamma\left(\dfrac{d}{2}\right)\left(2\pi\right)^d}\int_0^\infty dk \dfrac{k^{d-1}\sinh(\beta_cmc^2)}{\cosh\left(\beta_c\sqrt{\hbar^2c^2k^2-m^2c^4}\right)}-\cosh\left(\beta_cmc^2\right)\]

Case 1. Low T, with k_BT<<mc^2. Then T_C(B\overline{B},NR)=T_c(NR,B), as we would expect.

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

(7)   \begin{equation*} \boxed{k_BT_c(UR,B\overline{B})=\left[\dfrac{\hbar^dc^{d-2}\Gamma\left(\dfrac{d}{2}\right)(2\pi)^d}{4m\pi^{d/2}\Gamma(d)\zeta(d-1)}\right]^{1/(d-1)}n^{1/(d-1)}} \end{equation*}

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

    \[k_BT(UR,B\overline{B},3d,m)=\left(\dfrac{3\hbar^3c}{m}\right)^{1/2}n^{1/2}\]

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

    \[n=n_0+\dfrac{\Omega_d}{(2\pi \hbar)^d}\int_{0^+}^\infty \dfrac{1}{e^{\beta(E_p-\mu)}-1}p^{d-1}dp\]

    \[n_0\equiv\dfrac{1}{V\left[e^{\beta(mc^2-\mu)}-1\right]}=\dfrac{N_0}{L^d}\]

    \[\dfrac{U^B}{V}=u_B=nmc^2+\dfrac{\Omega_d}{(2\pi \hbar)^d}\int_{0^+}^\infty\dfrac{E_p-mc^2}{e^{\beta(E_p-\mu)}-1}p^{d-1}dp\]

    \[\dfrac{F(T,V,n)}{V}=\mathcal{F}=nmc^2+k_BT\dfrac{\Omega_d}{(2\pi \hbar)^d}\int_{0^+}^\infty \ln\left[1- e^{\beta(E_p-\mu)}\right]p^{d-1}dp\]

    \[\dfrac{S^B}{V}=s_B=\dfrac{k_B\Omega_d}{(2\pi \hbar)^d}\int_{0^+}^\infty p^{d-1}dp\dfrac{(E_p-mc^2)\beta}{e^{\beta(E_p-\mu)}-1}-\ln\left(1-e^{\beta(\mu-E_p)}\right)\]

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

    \[n=n_0+\dfrac{\Omega_d}{(2\pi \hbar)^d}\int_{0^+}^\infty dpp^{d-1} \dfrac{1}{e^{\beta(E_p-\mu)}-1}+\dfrac{1}{e^{\beta(E_p+\mu)}-1}\]

    \[\dfrac{U^{B\overline{B}}}{V}=u_{B\overline{B}}=nmc^2+\dfrac{\Omega_d}{(2\pi \hbar)^d}\int_{0^+}^\infty dpp^{d-1} \dfrac{E_p-mc^2}{e^{\beta(E_p-\mu)}-1}+\dfrac{E_p+mc^2}{e^{\beta(E_p+\mu)}-1}\]

    \[\dfrac{F(T,V,n)_{B\overline{B}}}{V}=\mathcal{F}=nmc^2+\dfrac{k_BT\Omega_d}{(2\pi \hbar)^d}\int_{0^+}^\infty dpp^{d-1}\{ \ln\left[1- e^{\beta(mc^2-E)}\right]+\ln\left[1- e^{-\beta(mc^2+E)}\right]\}\]

    \[\begin{split}\dfrac{S^{B\overline{B}}}{V}=s_{B\overline{B}}=\dfrac{k_B\Omega_d}{(2\pi \hbar)^d}\int_{0^+}^\infty dpp^{d-1}\{\dfrac{(E_p-mc^2)\beta}{e^{\beta(E_p-mc^2)}-1}+\dfrac{(E_p+mc^2)\beta}{e^{\beta(E_p+mc^2)}-1}+\\-\ln\left(1-e^{\beta(mc^2-E_p)}\right)-\ln\left(1-e^{-\beta(mc^2+E_p)}\right)\}\end{split}\]

Challenge for eager readers:

Take the density of states

    \[\rho(\varepsilon)=\dfrac{2\pi^{d/2}}{(2\pi)^{d/2}\Gamma\left(\dfrac{d}{2}\right)}\varepsilon\left(\varepsilon^2-m^2\right)^{(d-2)/2}\]

and take the UR limit to get and prove

    \[\rho(\varepsilon)=\dfrac{2\pi^{d/2}\varepsilon^{d-1}}{(2\pi)^d\Gamma\left(\dfrac{d}{2}\right)}\]

and calculate/check the critical temperature

    \[T_c=\left[\dfrac{(2\pi)^d\Gamma\left(\dfrac{d}{2}\right)q}{4\pi^{d/2}\Gamma(d)\zeta(d-1)m}\right]^{1/(d-1)}\]

obtaining the value of the “constant” q 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:

    \[\varepsilon(ideal)=\dfrac{U}{V_d}=\dfrac{2d\varepsilon_f k_f^d}{2^d\pi^{d/2}(d+2)\Gamma\left(\dfrac{d}{2}+1\right)}\]

Moreover, you have

    \[P(ideal)=\dfrac{4\varepsilon_f k_f^d}{2^d\pi^{d/2}(d+2)\Gamma\left(\dfrac{d}{2}+1\right)}\]

    \[n=\dfrac{2k_f^d}{2^d\pi^{d/2}\Gamma\left(\dfrac{d}{2}+1\right)}\]

and the Fermi energy reads (\hbar=c=1):

    \[E_f=\sqrt{k_f^2+m^2}=\sqrt{p_f^2+m^2}\]

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

    \[f_{sw}(NR)=\dfrac{d+2}{d}=\dfrac{D+1}{D-1}\]

    \[f_{sw}(UR)=\dfrac{d+1}{d}=\dfrac{D}{D-1}\]

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

    \[E=\dfrac{p^2_d}{2m}=\dfrac{p_1^2+\cdots+p_d^2}{2m}\]

and the energy in terms of Fermi quantities reads

    \[\boxed{\varepsilon=-\dfrac{2}{2^d\pi^{d/2}}\dfrac{\varepsilon_fk_f^d}{\Gamma\left(\dfrac{d}{2}+1\right)d}}\]

and the density of states

    \[g(\varepsilon)=\dfrac{g_sV2\pi^{d/2}p^{d-1}_r(\varepsilon)}{(2\pi\hbar)^d}\left(\dfrac{\partial \varepsilon}{\partial p_r}\right)^{-1}\]

Thus, the energy density will be

    \[\varepsilon=\dfrac{E}{V_d}=\dfrac{g_s}{\lambda_T^2}\dfrac{\beta^{d/2}E^{d/2}-1}{\Gamma\left(\dfrac{d}{2}\right)}\]

with the thermal wavelength

    \[\lambda_T=\dfrac{2\pi \hbar}{\sqrt{2m\pi \beta^{-1}}}\]

Furthermore

    \[N=\dfrac{g_sV}{\lambda_T^d}f_{d/2}(z)\]

where z=\exp(\mu \beta) is the fugacity. The Fermi function reads

    \[f_{d/2}(z)=\sum_{n=1}^\infty(-1)^{n+1}\dfrac{z^n}{k^{d/2}}=-Li_{d/2}(-z)\]

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

    \[\langle E\rangle_d=\dfrac{\langle E\rangle_d}{fermion}=\dfrac{\int_0^{E_F}g_d(E)F(E)dE}{\int_0^{E_f}g_d(E)F(E)dE}=\dfrac{d}{d+2}E_F\]

You get \langle E\rangle=E_F/2 if d=2, \langle E\rangle=3E_F/5 if d=3 and you also have

    \[\lim_{n\rightarrow\infty}\langle E\rangle_d=E_F\]

with

    \[g_d(E)dE=C(m,V)E^{(d-2)/2}dE\]

    \[E=\dfrac{p_1^2+\cdots+p_d^2}{2m}\]

    \[d^dp=\left(\dfrac{d}{2}\right)\dfrac{\pi^{d/2}(2m)^{d/2}\varepsilon^{(d-2)/2}d\varepsilon}{\Gamma\left(\dfrac{d}{2}+1\right)}\]

and the number density

    \[n=\dfrac{2\pi^{d/2}(2m)^{d/2}\varepsilon_f^{d/2}}{h^d\Gamma\left(\dfrac{d}{2}+1\right)}\]

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

    \[\boxed{\mathcal{F}=-\dfrac{1}{\beta^{d+1}}\dfrac{\Gamma\left(d+1\right)\zeta\left(d+1\right)}{2^{d-1}\pi^{d/2}\Gamma\left(\dfrac{d}{2}\right)d}}\]

The Casimir energy of such a bosonic field requires

    \[\left(\sum_{n\in Z}\vert n\vert^d\right)_{reg}=2\zeta(-d)\]

and the regularized energy in vacuum has to be

    \[\varepsilon_0(reg)=-\dfrac{1}{\beta^{d+1}}\pi^{d/2}\Gamma\left(-\dfrac{d}{2}\right)\zeta\left(-d\right)\]

and it shows that the Riemann zeta functional equation \xi(\nu)=\xi(1-\nu) holds iff

    \[\varepsilon_0=\mathcal{F}\]

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).

multiDobjectmultiDworldsheet

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,

    \[\phi=E(r)S_{d-1}=Q\]

up to a multiplicative constant, related to K_C=\dfrac{1}{4\pi \varepsilon_0} in D=4 spacetime. There, E(r) is the radial electric field, S_{d-1} is certain gaussian hypersurface containing the electric charge Q. If we take a hypersphere, so S_{d-1} equals the surface of the (d-1)-sphere, we would obtain a Coulomb-like law in D=d-1 dimensions:

(1)   \begin{equation*} f_C=\dfrac{\Gamma\left(\dfrac{d}{2}\right)}{2\pi^{d/2}}\dfrac{1}{r^{d-1}} \end{equation*}

Note that we have used units in which K_C=1 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 V(r) through the usual definition

    \[-\dfrac{\partial V(r)}{\partial r}=E(r)\]

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)   \begin{equation*} V(r)=\dfrac{(M_Xr)^{d/2-1}K_{d/2-1}(M_Xr)}{(2\pi)^{d/2}}\dfrac{1}{r^{d/2}} \end{equation*}

and where K_n(x) 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

    \[\int d^nx\exp(-\vec{x}^2)=\left(\int dx \exp(-x^2)\right)^n=\left(\Gamma(\dfrac{1}{2})\right)^n=\pi^{n/2}\]

    \[\int d^nx\exp (-\vec{x}^2)=\int dr r^{n-1}\exp(-r^2)S_{n-1}\]

and thus

    \[\boxed{S_{n-1}=\dfrac{\Gamma^n\left(\dfrac{1}{2}\right)}{\dfrac{1}{2}\Gamma\left(\dfrac{n}{2}\right)}=\dfrac{2\pi^{n/2}}{\Gamma\left(\dfrac{n}{2}\right)}}\]

The volume for the hypersphere is

    \[V_{d-1}=\dfrac{2\pi^{d/2}r^{d-1}}{\Gamma\left(\dfrac{d}{2}\right)}\]

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

    \[\sum_{i=1}^dx_i^2=r^2\]

and the solid sphere is instead the constrain

    \[\sum_{i=1}^dx_i^2\leq r^2\]

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

(3)   \begin{equation*} E(r)=\dfrac{\Gamma\left(\dfrac{d}{2}\right)Q}{2\pi^{d/2}}\dfrac{Q\vec{u_r}}{\varepsilon_{0,(d+1)} r^{d-1}} \end{equation*}

By dimensional analysis, we get that

    \[\left[\varepsilon_{0,(d+1)}\right]=\left[\varepsilon_{0,\cdot (d+1)}\right]=\dfrac{\left[\varepsilon_0\right]}{L^{d-3}}\]

Moreover, you have

    \[\varepsilon_{0,(d+1)}=\dfrac{\varepsilon_0}{\left(2R\sqrt{\pi}\right)^{d-3}\Gamma\left(\dfrac{d-3}{2}\right)}\]

to be more precise for any d>3.

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

(4)   \begin{equation*} F(q_1\rightarrow q_2;D)=\dfrac{\Gamma\left(\dfrac{d}{2}\right)q_1q_2\vec{u_r}}{2\pi^{d/2}\varepsilon_{0,(d+1)}r^{d-1}} \end{equation*}

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

(5)   \begin{equation*} E(r)=\dfrac{\Gamma\left(\dfrac{d}{2}\right)}{2\pi^{d/2}}\dfrac{1}{\varepsilon_{0,(d+1)}}\int_0^{q'}\dfrac{\vec{u_r}(21)dq'}{r^{d-1}} \end{equation*}

(6)   \begin{equation*} V(r)=\dfrac{\Gamma\left(\dfrac{d}{2}\right)}{2\pi^{d/2}\cdot (d-2)}\dfrac{1}{\varepsilon_{0,(d+1)}}\int_0^{q'}\dfrac{dq'}{r^{d-2}} \end{equation*}

 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)   \begin{equation*} B(r)=\dfrac{\Gamma\left(\dfrac{d-1}{2}\right)}{2\pi^{(d-1)/2}}\dfrac{\mu_{d+1}I_0}{r^{d-2}} \end{equation*}

A dimensional check provides that

    \[\left[\mu_{d+1}\right]=\dfrac{\left[\mu_0\right]}{\left[L^{3-d}\right]}\]

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

    \[\sqrt{\mu_{d+1,0}\varepsilon_{d+1,0}}=LT^{-1}\]

For one steady current I_1 over other I_2 parallel (both filamentary) the magnetic force per unit length equals to

(8)   \begin{equation*} \dfrac{F(I_1\rightarrow I_2;D)}{L}=-\dfrac{\Gamma\left(\dfrac{d-1}{2}\right)\mu_{d+1}I_1I_2}{r_{21}^{d-2}}\vec{u_r}(21) \end{equation*}

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)   \begin{equation*} B_{IJ}=\dfrac{\Gamma\left(\dfrac{d-1}{2}\right)}{4\pi^{(d-1)/2}}\mu_{d+1}\int\dfrac{J_1dA_{IJ}}{r^3} \end{equation*}

For a two dimensional cross section current

(10)   \begin{equation*} B_{IJ}=\dfrac{\Gamma\left(\dfrac{d-1}{2}\right)}{4\pi^{(d-1)/2}}\mu_{d+1}\int\dfrac{J_2dA_{IJ}ds}{r^3} \end{equation*}

and for the three dimensional cross section current

(11)   \begin{equation*} B_{IJ}=\dfrac{\Gamma\left(\dfrac{d-1}{2}\right)}{4\pi^{(d-1)/2}}\mu_{d+1}\int\dfrac{J_3\left(R_JdV_I-R_IdV_J\right)ds}{r^3} \end{equation*}

and where J_1,J_2,J_3 are current densities, and d\vec{s}=ds is along the current I.

Remark: In the 3+1 world we recover the classical and common result f_C\propto r^{-2} and

    \[f_C=\dfrac{K_CQq}{r^2}=\dfrac{Qq}{4\pi\varepsilon_0r^2}\]

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

    \[\nabla^2V(r)=0\]

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

    \[V(r)=\int d^dk\dfrac{e^{ik\cdot x}}{(2\pi)^dk^2}\]

and it implies an integral representation

    \[V(r)=\int \dfrac{dk}{(2\pi)^d}k^{d-3}d^{d-1}\Omega e^{ikr\cos\theta_1}\]

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

    \[(\theta_1,\theta_2,\ldots,\theta_{d-2},\varphi)\]

so we have

    \[x_1=r\cos\theta_1\]

    \[x_2=r\sin\theta_1\cos\theta_2\]

    \[x_D=r\sin\theta_1\sin\theta_2\cdots\sin\theta_{d-2}\cos\varphi\]

and

    \[d^{d-1}\Omega=\sin^{d-2}\theta_1\sin^{d-3}\theta_2\cdots \sin\theta_{d-2}d\theta_1d\theta_2\cdots d\theta_{d-2}d\varphi\]

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

    \[I_d=\int_0^\infty d\mu\sqrt{\pi}\Gamma\left(\dfrac{d-1}{2}\right)\dfrac{_0F_1\left(\frac{d}{2},-\frac{\mu^2}{4}\right)}{\Gamma\left(\dfrac{d}{2}\right)}\mu^{d-3}=2^{d-3}\sqrt{\pi}\Gamma\left(\dfrac{d-2}{2}\right)\Gamma\left(\dfrac{d-1}{2}\right)\]

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

    \[V(r)=\dfrac{\Gamma\left(\dfrac{d-2}{2}\right)}{4\pi^{d/2}}\dfrac{1}{r^{d-2}}\]

Q.E.D.

LOG#165. Rogers-Ramanujan identities.

r_rramanujancfformules Zeta impairs

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)   \begin{equation*} G(q)=\sum_{n=0}^\infty\dfrac{q^{n^2}}{(q;q)_n}=\dfrac{1}{(q;q^5)_\infty(q^4;q^5)_\infty}=1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots \end{equation*}

(2)   \begin{equation*} H(q)=\sum_{n=0}^\infty\dfrac{q^{n^2+n}}{(q;q)_n}=\dfrac{1}{(q^2;q^5)_\infty(q^3;q^5)_\infty}=1+q^2+q^3+q^4+q^5+2q^6+\cdots \end{equation*}

and where q=\exp(2\pi i\tau). It shows that q^{-1/60}G(q) and q^{11/60}H(q) are modular functions of \tau. 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)   \begin{equation*} 1+\frac{q}{1+\frac{q^2}{1+\frac{q^3}{1+\cdots}}}  = \frac{G(q)}{H(q)} \end{equation*}

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.

mockthetaOrderTwo

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)   \begin{equation*} \vartheta(z;\tau)=\sum_{n=-\infty}^\infty e^{\pi i n^2\tau+2\pi i n z}=\sum_{n=-\infty}^\infty q^{n^2}\eta^n=1+2\sum_{n=1}^\infty\left(e^{\pi i \tau}\right)^{q^2}\cos (2\pi nz) \end{equation*}

where q=e^{i\pi \tau} and \eta=e^{2\pi i z}. It satisfies the functional equation

(2)   \begin{equation*} \vartheta(z+1;q)=\vartheta(z;\tau) \end{equation*}

and

(3)   \begin{equation*} \vartheta(z+a+b\tau;\tau)=e^{-\pi ib^2\tau-2\pi i bz}\vartheta(z;\tau) \end{equation*}

\forall a,b\in Z. The Jacobi theta function is related to the Riemann zeta function. With

    \[\vartheta(0;-\dfrac{1}{\tau})=(-i\tau)^{1/2}\vartheta(0;\tau)\]

we have

(4)   \begin{equation*} \Gamma(\frac{s}{2})\pi^{-s/2}\zeta(s)=\dfrac{1}{2}\int_0^\infty\left[\vartheta(0;it)-1\right]t^{s/2}\dfrac{dt}{2} \end{equation*}

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

    \[\vartheta_{00}(z;\tau)=\vartheta(z;\tau)\]

    \[\vartheta_{01}(z;\tau)=\vartheta(z+\frac{1}{2};\tau)\]

    \[\vartheta_{10}(z;\tau)=e^{\pi i \tau/4+\pi iz}\vartheta(z+\frac{1}{2};\tau)\]

    \[\vartheta_{11}(z;\tau)=e^{\pi i \tau/4+\pi i(z+1/2)}\vartheta(z+\frac{1}{2}\tau+\frac{1}{2};\tau)\]

    \[\vartheta_{1}(z;q)=-\vartheta_{11}(z;\tau)\]

    \[\vartheta_{2}(z;q)=+\vartheta_{10}(z;\tau)\]

    \[\vartheta_{3}(z;q)=+\vartheta_{00}(z;\tau)\]

    \[\vartheta_{4}(z;q)=+\vartheta_{01}(z;\tau)\]

where q=e^{i\pi\tau} is sometimes called the nome. Auxiliary theta functions can be written in terms of the nome, and if

    \[\alpha=(-i\tau)^{1/2}e^{\pi i z^2/\tau}\]

they satisfy the following Jacobi identities

    \[\vartheta_{00}\left(\dfrac{z}{\tau};-\dfrac{1}{\tau}\right)=\alpha\vartheta_{00}(z;\tau)\]

    \[\vartheta_{10}\left(\dfrac{z}{\tau};-\dfrac{1}{\tau}\right)=\alpha\vartheta_{01}(z;\tau)\]

    \[\vartheta_{01}\left(\dfrac{z}{\tau};-\dfrac{1}{\tau}\right)=\alpha\vartheta_{10}(z;\tau)\]

    \[\vartheta_{11}\left(\dfrac{z}{\tau};-\dfrac{1}{\tau}\right)=-i\alpha\vartheta_{11}(z;\tau)\]

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

    \[\vartheta(x;it)=1+2\sum_{n=1}^\infty\exp (-\pi n^2t)\cos(2\pi n x)\]

we find that it verifies

    \[\partial_t\vartheta(x;it)=\dfrac{1}{4\pi}\partial_{xx}\vartheta(x;it)\]

with

    \[\lim_{n\rightarrow 0}\vartheta(x;it)=\sum_{n=-\infty}^\infty\delta(x-n)\]

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:

    \[\varphi(e^{-\pi x}) = \vartheta(0; {\mathrm{i}}x) = \theta_3(0;e^{-\pi x}) = \sum_{n=-\infty}^\infty e^{-x \pi n^2}\]

    \[\varphi\left(e^{-\pi} \right) = \frac{\sqrt[4]{\pi}}{\Gamma(\frac{3}{4})}\]

    \[\varphi\left(e^{-2\pi} \right) = \frac{\sqrt[4]{6\pi+4\sqrt2\pi}}{2\Gamma(\frac{3}{4})}\]

    \[\varphi\left(e^{-3\pi}\right) = \frac{\sqrt[4]{27\pi+18\sqrt3\pi}}{3\Gamma(\frac{3}{4})}\]

    \[\varphi\left(e^{-4\pi}\right) =\frac{\sqrt[4]{8\pi}+2\sqrt[4]{\pi}}{4\Gamma(\frac{3}{4})}\]

    \[\varphi\left(e^{-5\pi} \right) =\frac{\sqrt[4]{225\pi+ 100\sqrt5 \pi}}{5\Gamma(\frac{3}{4})}\]

    \[\varphi\left(e^{-6\pi}\right) = \frac{\sqrt[3]{3\sqrt{2}+3\sqrt[4]{3}+2\sqrt{3}-\sqrt[4]{27}+\sqrt[4]{1728}-4}\cdot \sqrt[8]{243{\pi}^2}}{6\sqrt[6]{1+\sqrt6-\sqrt2-\sqrt3}{\Gamma(\frac{3}{4})}}\]

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

    \[\theta_F=\sum_{m\in Z^n}e^{2\pi i z F(m)}\]

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

(5)   \begin{equation*} \theta(z;\tau)=\sum_{m\in Z^n}e^{2\pi i\left(\frac{1}{2}m^T\tau m+m^Tz\right)} \end{equation*}

and where z\in C^n is a n-dimensional complex vector, T denotes the transpose, \tau\in H_n, such as H_n=\{F\in Mat_n(C)s.t.F=F^T,ImF>0\}. 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 Sp_{2n}(R) is infinite cyclic, so it has a unique connected double cover, which is denoted Mp_{2}(R) and it denotes the metaplectic group.

The metaplectic group Mp_{2}(R) 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)   \begin{equation*} \theta(z+a+\tau b;\tau)=e^{2\pi i(-b^Tz-\frac{1}{2}b^T\tau b)}\theta(z;\tau) \end{equation*}

\forall a,b\in Z^n, \forall z\in C^n,\forall \tau \in H_n

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.

RamanujanSeal

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)   \begin{equation*} (a;q)_n=\prod_{k=0}^{n-1}(1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1}) \end{equation*}

with (a;q)_0\equiv 1. The infinite product extension is also very popular

(2)   \begin{equation*} \boxed{(a;q)_\infty=\prod_{k=0}^\infty(1-aq^k)} \end{equation*}

and it is analytic in the unit disc, with \phi(q)=(q;q)_\infty being the Euler’s function, important object in combinatorics, number theory and the theory of modular forms.

    \[\boxed{\phi(q)=(q;q)_\infty=\prod_{k=1}^\infty(1-q^k)}\]

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

(3)   \begin{equation*} \boxed{(a;q)_n=\dfrac{(a;q)_\infty}{(aq^n;q)_\infty}} \end{equation*}

(4)   \begin{equation*} \boxed{(a;q)_{-n}=\dfrac{1}{(aq^{-n};q)_n}=\prod_{k=1}^n\dfrac{q}{1-aq^{-k}}} \end{equation*}

(5)   \begin{equation*} \boxed{(a;q)_{-n}=\dfrac{(-q/a)^nq^{n(n-1)/2}}{(q/a;q)_n}} \end{equation*}

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

(6)   \begin{equation*} \boxed{\dfrac{(ax;q)_\infty}{(x;q)_\infty}=\sum_{n=-\infty}^\infty\dfrac{(a;q)_n}{(q;q)_n}x^n} \end{equation*}

Interpretation: the coefficient of q^ma^n in the expansion

    \[(a;q)^{-1}=\prod_{k=0}^\infty(1-aq^k)^{-1}=\sum_{k=0}^\infty\dfrac{a^k}{(q;q)_k}\]

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

    \[(-a;q)_\infty=\prod_{k=0}^\infty(1+aq^k)\]

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

    \[(a_1,a_2,\ldots,a_m;q)_n=(a_1;q)_n(a_2;q)_n\ldots(a_m;q)_n\]

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

    \[\left[n\right]_q=\dfrac{1-q^n}{1-q}\]

and the q-factorial is

    \[\left[n_q\right]!=\prod_{k=1}^n\left[k\right]_q=\left[1\right]_q\left[2\right]_q\cdots\left[n-1\right]_q\left[n\right]_q=\dfrac{(q;q)_n}{(1-q)^n}\]

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

(7)   \begin{equation*} \left(\dfrac{d}{dx}\right)_qf(x)=D_qf(x)=\dfrac{f(qx)-f(x)}{qx-x} \end{equation*}

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

    \[D_q(f(x)+g(x))=D_qf(x)+D_qg(x)\]

    \[D_q(f\cdot g)(x)=g(x)D_qf(x)+f(qx)D_qg(x)=g(qx)D_qf(x)+f(x)D_qg(x)\]

Moreover, we also have

    \[D_q^nf(0)=\dfrac{f^{(n)}(0)}{n!}\dfrac{(q;q)_n}{(1-q)^n}=\dfrac{f^{(n)}(0)}{n!}\left[n\right]_q\]

The Taylor expansion analogue also exists:

    \[f(z)=\sum_{n=0}f^{(n)}(0)\dfrac{z^n}{n!}=\sum_{n=0}^\infty (D_q^nf)(0)\dfrac{z^n}{\left[n\right]_q!}\]

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

(8)   \begin{equation*} \theta(z;q)=\prod_{n=0}^\infty(1-q^nz)\left(1-\frac{q^{n+1}}{z}\right) \end{equation*}

and with 0\leq \vert q\vert <1 this yields

(9)   \begin{equation*} \theta(z;q)=\theta\left(\dfrac{q}{z};q\right)=-z\theta\left(\dfrac{1}{z};q\right)=\dfrac{(z;q)_\infty}{\left(\dfrac{q}{z};q\right)_\infty} \end{equation*}

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

(10)   \begin{equation*} \boxed{f(a,b)=\sum_{n=-\infty}^\infty a^{n(n+1)/2}b^{n(n-1)/2}} \end{equation*}

with \vert ab\vert <1. 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

    \[\boxed{f(a,b)=(-a;ab)_\infty(-b;ab)_\infty(ab;ab)_\infty}\]

Some additional identities of this Ramanujan theta function are:

(11)   \begin{equation*} f(q;q)=\sum_{n=-\infty}^\infty q^{n^2}=(-q;q^2)_\infty^2(q^2;q^2)_\infty \end{equation*}

(12)   \begin{equation*} f(q;q^3)=\sum_{n=-\infty}^\infty q^{n(n+1)/2}=(q^2;q^2)_\infty(-q;q)_\infty \end{equation*}

and

(13)   \begin{equation*} f(-q;-q^2)=\sum_{n=-\infty}^\infty(-1)^nq^{n(3n-1)/2}=(q;q)_\infty=\phi(q) \end{equation*}

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

    \[\boxed{\vartheta(w;q)=f(wq^2;qw^{-2})}\]

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

    \[\prod_{m=1}^\infty\left( 1 - x^{2m}\right)\left( 1 + x^{2m-1} y^2\right)\left(1 +\frac{x^{2m-1}}{y^2}\right)= \sum_{n=-\infty}^\infty x^{n^2} y^{2n}\]

and it is defined for arbitrary complex numbers x,y such as \vert x\vert<1 and y\neq 0. 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

    \[\sum_{n=-\infty}^\infty q^{\frac{n(n+1)}{2}}z^n=(q;q)_\infty\left(-\frac{1}{z};q\right)_\infty (-zq;q)_\infty\]

where (a;q)_\infty is the infinite q-Pochhammer symbol and

    \[\sum_{n=-\infty}^\infty a^{\frac{n(n+1)}{2}}b^{\frac{n(n-1)}{2}}= (-a; ab)_\infty (-b; ab)_\infty (ab;ab)_\infty\]

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).

polylog

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

Firstly, \forall s\in C\neq 0,1,2,3 we have

(1)   \begin{equation*} Li_s(e^\mu)=\dfrac{\Gamma (1-s)}{(2\pi)^{1-s}}\left[i^{1-s}\zeta (1-s,\dfrac{\mu}{2\pi i})+i^{s-1}\zeta (1-s,1-\dfrac{\mu}{2\pi i})\right] \end{equation*}

and now, if 0<Im(\mu)\leq 2\pi

(2)   \begin{equation*} Li_s(e^\mu)=\Gamma(1-s)\left[(-2\pi i)^{s-1}\sum_{k=0}^\infty\left(k+\dfrac{\mu}{2\pi i}\right)^{s-1}+(2\pi i)^s\sum_{k=0}^\infty\left(k+1-\dfrac{\mu}{2\pi i}\right)^{s-1}\right] \end{equation*}

(3)   \begin{equation*} Li_s(e^\mu)=\Gamma (1-s)\sum_{k=-\infty}^\infty\left(2\pi i k-\mu\right)^{s-1} \end{equation*}

\forall Re(s)<0,\forall \mu/e^\mu\neq 1

The next identity also holds

    \[Li_s(e^\mu)=\Gamma (1-s)(-\mu)^{s-1}+\sum_{k=0}^\infty\dfrac{\zeta (s-k)\mu^k}{k!}\]

The Bose-Einstein integral can be rewritten for Re(s)>0 as follows

    \[Li_s(z)=\dfrac{z}{2}+\dfrac{z}{2\Gamma (s)}\int_0^\infty e^{-t}t^{s-1}\coth\left(\dfrac{t-\ln z}{2}\right)dt\]

and

    \[Li_s(z)=\dfrac{z}{2}+\sum_{k=-\infty}^\infty\dfrac{\Gamma (1-s,2\pi ik-\ln z)}{(2\pi ik-\ln z)^{1-s}}\]

The polylog has the following asymptotic series. If \vert z\vert >>1 we have

(4)   \begin{equation*} Li_s(z)=\dfrac{\pm iz}{\Gamma (s)}\left[\ln (-z)\pm i\pi\right]^{s-1}-\sum_{k=0}^\infty(-1)^k(2\pi)^{2k}\dfrac{B_{2k}}{(2k)!}\dfrac{\left[\ln(-z)\pm i\pi\right]^{s-2k}}{\Gamma (s+1-2k)} \end{equation*}

(5)   \begin{equation*} Li_s(z)=\sum_{k=0}^\infty(-1)^k(1-2^{1-2k})(2\pi)^{2k}\dfrac{B_{2k}}{(2k)!}\dfrac{\left[\ln(-z)\right]^{s-2k}}{\Gamma (s+1-2k)} \end{equation*}

Euler’s dilogarithm or Spence’s function has nice features. Interestingly, computer algebra systems generally define the dilogarithm as Li_2(1-z), thus be aware with the definitions you read, use and write. In our case:

(6)   \begin{equation*} Li_2(z)=-\int_0^z\dfrac{\ln (1-t)}{t}dt=-\int_0^1\dfrac{\ln (1-zt)}{t}dt \end{equation*}

Moreover, for Re(z)\geq 1 you can find

(7)   \begin{equation*} Li_2(z)=\dfrac{\pi^2}{6}-\int_1^z\dfrac{\ln (t-1)}{t}dt-i\pi \ln (z) \end{equation*}

(8)   \begin{equation*} Li_2(z)=\dfrac{\pi^2}{3}-\dfrac{1}{2}\left(\ln (z)\right)^2-\sum_{k=1}^\infty\dfrac{1}{k^2z^k}-i\pi\ln z \end{equation*}

\forall x\notin (1,\infty) and \notin (1,\infty) we have the Abel identity:

(9)   \begin{equation*} \ln(1-x)\ln(1-y)=Li_2\left(\dfrac{x}{1-y}\right)+Li_2\left(\dfrac{y}{1-x}\right)-Li_2(x)-Li_2(y)-Li_2\left(\dfrac{xy}{(1-x)(1-y)}\right) \end{equation*}

Euler’s reflection formula with y=1-x follows up

(10)   \begin{equation*} Li_2(x)+Li_2(1-x)=\dfrac{\pi^2}{6}-\ln (x)\ln(1-x) \end{equation*}

    \[Li_2(1)=\zeta (2)=\dfrac{\pi^2}{6}\]

The pentagon identity, with u=\dfrac{x}{1-y} and v=\dfrac{y}{1-x} is a mutation of the Abel identity

    \[\boxed{Li_2(u)+Li_2(v)-Li_2(uv)=Li_2\left(\dfrac{u-uv}{1-uv}\right)+Li_2\left(\dfrac{v-uv}{1-uv}\right)+\ln\left(\dfrac{1-u}{1-uv}\right)\ln\left(\dfrac{1-v}{1-uv}\right)}\]

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

    \[Li_2(-x)=-Li_2\left(\dfrac{x}{1+x}\right)-\dfrac{1}{2}\left[\ln\left(x+1\right)\right]^2\]

with x\notin (-\infty,-1). We have an inversion formula for this too

    \[Li_2(x)+Li_2\left(\dfrac{1}{x}\right)=-\dfrac{\pi^2}{6}-\dfrac{1}{2}\left(\ln(-x)\right)^2\]

with x\notin (0,1), or

    \[Li_2(x)+Li_2\left(\dfrac{1}{x}\right)=\dfrac{\pi^2}{3}-\dfrac{1}{2}\left(\ln x\right)^2-i\pi\ln x\]

with x\geq 1.

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:

    \[Li_2(-1)=-\dfrac{\pi^2}{12}\]

    \[Li_2(0)=0\]

    \[Li_2(1/2)=\dfrac{\pi^2}{12}-\dfrac{\ln^22}{2}\]

    \[Li_2(1)=\dfrac{\pi^2}{6}=\zeta(2)\]

    \[Li_2(2)=\dfrac{\pi^2}{4}-i\pi\ln 2\]

Moreover, suppose that x=-\phi=-\dfrac{1+\sqrt{5}}{2} is the negative golden mean ratio, then

    \[Li_2(-\phi)=-\dfrac{\pi^2}{10}-\ln^2\left(\dfrac{\sqrt{5}-1}{2}\right)\]

If x=\phi-1=1/\phi we also have

    \[Li_2(1/\phi)=Li_2\left(\phi-1\right)=-\dfrac{\pi^2}{15}+\dfrac{1}{2}\ln^2\left(\dfrac{\sqrt{5}-1}{2}\right)\]

If x=-\dfrac{1}{2}\left(3-\sqrt{5}\right)\equiv\Xi, we have

    \[Li_2(\Xi)=\dfrac{\pi^2}{15}-\ln^2\left(\dfrac{\sqrt{5}-1}{2}\right)\]

and

    \[Li_2(1-\phi)=\dfrac{\pi^2}{10}-\ln^2\left(\dfrac{\sqrt{5}-1}{2}\right)\]

    \[Li_2(\phi)=\dfrac{11\pi^2}{15}+\dfrac{1}{2}\ln^2\left(-\left(\dfrac{\sqrt{5}-1}{2}\right)\right)\]

    \[Li_2\left(\dfrac{\sqrt{5}+3}{2}\right)=-\dfrac{11\pi^2}{15}-\ln^2\left(-\left(\dfrac{\sqrt{5}+1}{2}\right)\right)\]

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

    \[\rho=\dfrac{1}{2}\left(\sqrt{5}-1\right)=\phi-1=\dfrac{1}{\phi}\]

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

(11)   \begin{equation*} Li_2(\rho^6)=4Li_2(\rho^3)+3Li_2(\rho^2)-6Li_2(\rho)+\dfrac{7}{30}\pi^2 \end{equation*}

and Landen also derived

(12)   \begin{equation*} Li_2(\rho)=\dfrac{\pi^2}{10}-\ln^2(\rho) \end{equation*}

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

(13)   \begin{equation*} 2^{1-s}Li_s(z^2)=Li_s(z)+Li_s(-z) \end{equation*}

Gauss wrote the next sum, a discrete Fourier transform

(14)   \begin{equation*} k^{1-s}Li_s(z^k)=\sum_{n=0}^{k-1}Li_s\left(ze^{2\pi in/k}\right) \end{equation*}

The Kummer’s identity for duplication reads

(15)   \begin{equation*} 2^{1-n}\Lambda_n(-z^2)=\Lambda_n(z)+\Lambda_n(-z) \end{equation*}

Moreover, if Char F=0, we can derive the result

    \[\lambda^{-\nu}J_\nu(\lambda z)=\sum_{n=0}^\infty\dfrac{1}{n!}\left(\dfrac{(1-\lambda^2)z}{2}\right)^nJ_{\nu+n}(z)\]

and where J_{\nu}(z) is the Bessel function with \lambda,\nu\in C.

Periodic zeta functions are defined by

(16)   \begin{equation*} F(s,q)=\sum_{m=1}^\infty\dfrac{e^{2\pi imq}}{m^s}=Li_s\left(e^{2 \pi i q}\right) \end{equation*}

and thus

    \[2^{-s}F(s,q)-F(s,\frac{q}{2})+F(s,\frac{q+1}{2})\]

    \[k^{-s}F(s,kq)=\sum_{n=0}^{k-1}F(s,q+\frac{n}{k})\]

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

    \[k^s\zeta(s)=\sum_{n=1}^k\zeta(s,\frac{n}{k})\]

    \[k^s\zeta(s,kz)=\sum_{n=0}^{k-1}\zeta (s,z+\frac{n}{k})\]

    \[\zeta(s,kz)=\sum_{n=0}^\infty\begin{pmatrix}s+n-1\\ n\end{pmatrix}(1-k)^nz^n\zeta(s+n,z)\]

See you in my next blog post!

LOG#161. Polylogia flashes(III).

poly_log_and_wood_siding_sealer_5477a3dab764b

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

(1)   \begin{equation*} Li_s(z)=\sum_{k=1}^\infty \dfrac{z^k}{k^s}=z+\dfrac{z^2}{2^s}+\dfrac{z^3}{3^s}+\cdots \forall z,\vert z\vert<1 \end{equation*}

and by analytic continuation that equation can be extended to all \vert z\vert>1. In fact

(2)   \begin{equation*} Li_{s+1}(z)=\int_0^z\dfrac{Li_s(t)}{t}dt \end{equation*}

such as Im(Li_s(z))=-\dfrac{\pi \mu^{s-1}}{\Gamma (s)} \forall z\geq 1, since we define

(3)   \begin{equation*} Im(Li_s(z+i\delta))=\dfrac{\pi \mu^{s-1}}{\Gamma (s)} \end{equation*}

and the equations

(4)   \begin{equation*} z\dfrac{\partial Li_s(z)}{\partial z}=Li_{s-1}(z) \end{equation*}

(5)   \begin{equation*} \dfrac{\partial Li_s(z)}{\partial \ln z}=Li_{s-1}(z) \end{equation*}

(6)   \begin{equation*} \dfrac{\partial Li_s(e^\mu)}{\partial \mu}=Li_{s-1}(e^\mu) \end{equation*}

Duplication formula for the polylogarithm:

(7)   \begin{equation*} \boxed{Li_s(-z)+Li_s(z)=2^{1-s}Li_s(z^2)} \end{equation*}

Connection with the Kummer’s function can be established

(8)   \begin{equation*} \Lambda_n(z)=\int_0^z\dfrac{\log^{n-1}\vert t\vert}{1+t}dt \end{equation*}

(9)   \begin{equation*} \Lambda_n(z)+\Lambda_n(-z)=2^{1-n}\Lambda_n(-z^2) \end{equation*}

and thus

(10)   \begin{equation*} Li_n(z)=Li_n(1)+\sum_{k=1}^{n-1}(-1)^{k-1}\dfrac{\log^k\vert z\vert}{k!}Li_{n-k}+\dfrac{(-1)^{n-1}}{(n-1)!}\left[\Lambda_n(-1)-\Lambda_n(-z)\right] \end{equation*}

We also have

(11)   \begin{equation*} \sum_{m=0}^{p-1}Li_s(ze^{2\pi i\frac{m}{p}})=p^{1-s}Li_s(z^p) \end{equation*}

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

(12)   \begin{equation*} Li_0(z)=\dfrac{z}{1-z} \end{equation*}

(13)   \begin{equation*} Li_1(z)=-\ln (1-z) \end{equation*}

(14)   \begin{equation*} Li_{-1}(z)=\dfrac{z}{(1-z)^2} \end{equation*}

(15)   \begin{equation*} Li_{-2}(z)=\dfrac{z(1+z)}{(1-z)^3} \end{equation*}

(16)   \begin{equation*} Li_{-3}(z)=\dfrac{z(1+4z+z^2)}{(1-z)^4} \end{equation*}

(17)   \begin{equation*} Li_{-4}(z)=\dfrac{z(1+z)(1+10z+z^2)}{(1-z)^5} \end{equation*}

And more generally, we have the general formulae

(18)   \begin{equation*} \boxed{Li_{-n}(z)=\left(z\dfrac{\partial}{\partial z}\right)^n\left(\dfrac{z}{1-z}\right)} \end{equation*}

(19)   \begin{equation*} Li_{-n}(z)=\left[\dfrac{\partial}{\partial \log (z)}\right]^n\left(\dfrac{z}{1-z}\right)=\sum_{k=0}^nk!S(n+1,k+1)\left(\dfrac{z}{1-z}\right)^{k+1} \end{equation*}

and where S(n,k) are the Stirling numbers of the second kind.

 

(20)   \begin{equation*} Li_{-n}(z)=(-1)^{n+1}\sum_{k=0}^nk!S(n+1,k+1)\left(-\dfrac{1}{1-z}\right)^{k+1} \end{equation*}

Furthermore,

(21)   \begin{equation*} Li_{-n}=\dfrac{1}{(1-z)^{n+1}}\sum_{k=0}^{n-1}\langle\begin{matrix}n\\ k\end{matrix}\rangle z^{n-k} \end{equation*}

and where \langle\begin{matrix}n\\ k\end{matrix}\rangle are the eulerian numbers.

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

    \[Li_1(\frac{1}{2})=\ln (2)\]

    \[Li_2(\frac{1}{2})=\dfrac{\pi^2}{12}-\dfrac{1}{2}\left(\ln 2\right)^2\]

    \[Li_3(\frac{1}{2})=\dfrac{1}{6}\ln^32-\dfrac{1}{12}\pi^2\ln (2)+\dfrac{7}{8}\zeta (3)\]

    \[Li_4(\frac{1}{2}))\dfrac{\pi^4}{360}-\dfrac{1}{24}\ln^42+\dfrac{\pi^2}{24}\ln^22-\dfrac{1}{2}\zeta(\overline{3},\overline{1})\]

and where

    \[\zeta(\overline{3},\overline{1})=\sum_{m>n>0}(-1)^{m+n}m^{-3}n^{-1}\]

    \[Li_s(e^{2\pi i\frac{m}{p}})=p^{-s}\sum_{k=1}^{p}e^{2\pi i m\frac{k}{p}}\zeta (s,\frac{k}{p})\]

with m=1,2,\ldots,p-1.

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

(22)   \begin{equation*} Li_s(1)=\zeta(s),Re(s)>1 \end{equation*}

(23)   \begin{equation*} Li_s(-1)=-\eta(s) \end{equation*}

(24)   \begin{equation*} Li_s(\pm i)=2^{-s}\eta(s)\pm i\beta(s) \end{equation*}

where \beta(s) is the Dirichlet beta function. The complete Fermi-Dirac integral is also polylogarithmic

    \[F_s(\mu)=-Li_{s+1}(-e^{\mu})\]

The incomplete polylog is also interesting

    \[Li_s(z)=Li_s(0,z)\]

    \[Li_s(b,z)=\dfrac{1}{\Gamma (z)}\int_b^{\infty}\dfrac{x^{s-1}dx}{\frac{e^x}{z}-1}\]

    \[Li_s(b,z)=\sum_{k=1}^\infty\dfrac{z^k}{k^s}\dfrac{\Gamma (s,kb)}{\Gamma (s)}\]

and the incomplete gamma function is defined by

    \[\Gamma (s,x)=\int_x^\infty t^{s-1}{e^{-t}}dt\]

    \[\gamma (s,x)=\int_0^xt^{s-1}e^{-t}dt\]

    \[Li_s(z)=z\Psi(z,s,1)\]

(25)   \begin{equation*} i^{-s}Li_s(e^{2\pi i x})+i^sLi_s(e^{-2\pi i x})=\dfrac{(2\pi)^s}{\Gamma(s)}\zeta (1-s,x) \end{equation*}

with 0\leq Re(x)<1, Im(x)\geq 0, 0<Re(x)\leq 1Im(x)<0. Moreover,

(26)   \begin{equation*} Li_s(z)=\dfrac{\Gamma (1-s)}{(2\pi)^{1-s}}\left[i^{1-s}\zeta\left((1-s,\dfrac{1}{2}+\dfrac{\ln(-z)}{2\pi i}\right)+i^{s-1}\zeta\left(1-s,\dfrac{1}{2}-\dfrac{\ln (-z)}{2\pi i}\right)\right] \end{equation*}

There is also a formula called inversion formula

(27)   \begin{equation*} Li_s(z)+(-1)^sLi_s(\frac{1}{z})=\dfrac{(2\pi i)^s}{\Gamma (s)}\zeta (1-s,\frac{1}{2}+\dfrac{\ln (-z)}{2\pi i} \end{equation*}

\forall s\in C and for z\notin (1,\infty)

(28)   \begin{equation*} Li_s(z)+(-1)^sLi_s\left(\dfrac{1}{z}\right)=\dfrac{(2\pi i)^s}{\Gamma (s)}\left(\zeta (1-s,\dfrac{1}{2}-\dfrac{\ln(-1/z)}{2\pi i}\right) \end{equation*}

The expression

    \[\zeta (1-n, x)=-\dfrac{B_n(x)}{n}\]

implies that

(29)   \begin{equation*} Li_n(e^{2\pi i x})+(-1)^nLi_n(e^{-2\pi i x})=-\dfrac{(2\pi i)^n}{n!}B_n(x) \end{equation*}

with

0\leq Re(x)<1 if Im(z)\geq 0 and 0<Re(x)\leq 1 if Im(x)<0. The following feynmanity (nullity) identity holds as well

(30)   \begin{equation*} \boxed{Li_{-n}(z)+(-1)^nLi_{-n}\left(\dfrac{1}{z}\right)=0} \end{equation*}

\forall n=1,2,3,\ldots and n=0,\pm 1,\pm 2,\ldots

(31)   \begin{equation*} Li_n(z)+(-1)^nLi_n(1/z)=-\dfrac{(2\pi i)^n}{n!}B_n\left(\dfrac{1}{2}+\dfrac{\ln (-z)}{2\pi i}\right) \end{equation*}

for z\notin (0,1) and

(32)   \begin{equation*} Li_n(z)+(-1)^nLi_n(1/z)=-\dfrac{(2\pi i)^n}{n!}B_n\left(\dfrac{1}{2}-\dfrac{\ln (-1/z)}{2\pi i}\right) \end{equation*}

for z\notin (1,\infty). Polylogs and Clausen functions are related (we already saw this before)

(33)   \begin{equation*} Li_s(e^{\pm i\theta})=Ci_s(\theta)\pm iSi_s(\theta) \end{equation*}

The inverse tangent integral is related to polylogs

(34)   \begin{equation*} Ti_s(z)=\dfrac{1}{2i}\left[Li_s(iz)-Li_s(-iz)\right] \end{equation*}

(35)   \begin{equation*} Ti_0(z)=\dfrac{z}{1+z^2} \end{equation*}

(36)   \begin{equation*} Ti_2(z)=\int_0^z\dfrac{\arctan(t)}{t}dt \end{equation*}

(37)   \begin{equation*} Ti_{n+1}(z)=\int_0^z\dfrac{Ti_n(t)}{t}dt \end{equation*}

The Legendre chi function \chi_s(z) is also related to polylogs

(38)   \begin{equation*} \chi_s(z)=\dfrac{1}{2}(Li_s(z)-Li_s(-z)) \end{equation*}

The incomplete zeta function or Debye functions are polylogs as well

(39)   \begin{equation*} Z_n(z)=\dfrac{1}{(n-1)!}\int_z^\infty \dfrac{t^{n-1}}{e^t-1}dt,\forall n=1,2,3,\ldots \end{equation*}

(40)   \begin{equation*} Li_n(e^\mu)=\sum_{k=0}^{n-1}Z_{n-k}(-\mu)\dfrac{\mu^k}{k!},\forall n=1,2,3,\ldots \end{equation*}

(41)   \begin{equation*} Z_n(z)=\sum_{k=0}^{n-1}Li_{n-k}(e^{-k})\dfrac{z^k}{k!},\forall n=1,2,3,\ldots \end{equation*}

Now, we will write some polylog integrals

(42)   \begin{equation*} Li_s(z)=\dfrac{1}{\Gamma (s)}\int_0^\infty\dfrac{t^{s-1}}{\frac{e^{t}}{z}-1}dt \end{equation*}

This last integral converges if Re(s)>0 \forall z\in R and z\geq 1. It is the Bose-Einstein distribution!

The Fermi-Dirac integrals read

(43)   \begin{equation*} -Li_s(-z)=\dfrac{1}{\Gamma (s)}\int_0^\infty\dfrac{t^{s-1}}{\frac{e^{t}}{z}+1}dt \end{equation*}

For Re(s)<0 and \forall z excepting z\in R and z\geq 0 we have

    \[Li_s(z)=\int_0^\infty\dfrac{t^{-s}\sin(s\pi/2-t\ln (-z))}{\sinh (\pi t)}dt\]

and

(44)   \begin{equation*} Li_s(e^{\mu})=-\dfrac{\Gamma (1-s)}{2\pi i}\oint_H\dfrac{(-t)^{s-1}}{e^{t-\mu}-1}dt \end{equation*}

with residue equal to

    \[R=\dfrac{i}{2\pi}=\Gamma(1-s)(-\mu)^{s-1}\]

See you in the next polylog post!

LOG#160. Polylogia flashes(II).

IntegerRelationsPolylog1q

The polylogarithm or Jonquière’s function is generally defined as

    \[\boxed{Li_s(z)=\sum_{n=1}^\infty \dfrac{z^n}{n^s}}\]

Do not confuse Li_s(z) with the logarithm integral in number theory, which is

    \[li(x)=\int_0^x\dfrac{dt}{\log t}\]

such as

    \[\int_0^x\dfrac{dt}{\ln t}\]

\forall 0<x<1 and

    \[V.P.\int_0^x\dfrac{dt}{\ln t}=\lim_{\varepsilon\rightarrow{0^+}}\left[\int_0^{1-\varepsilon}\dfrac{dt}{\ln t}+\int_{1+\varepsilon}^x\dfrac{dt}{\ln t}\right]\]

\forall x>1. In fact, notation can be confusing sometimes since the european li(x) is sometimes written as Li(s) and

    \[\int_0^1li(z)dz=-\ln(2)\]

    \[Li(x)=\int_2^x=li(x)-li(2)\]

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

    \[Li_s(z)=F(z,s)\]

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

(1)   \begin{equation*} \boxed{Li_s(z)=z\Phi(z,s,1)} \end{equation*}

 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)   \begin{equation*} \int_0^\infty \dfrac{k^s}{e^{k-\mu}+1}=-\Gamma(s+1)Li_{s+1}(-e^{\mu}) \end{equation*}

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

(3)   \begin{equation*} \int_0^\infty\dfrac{k^s}{e^{k-\mu}-1}=\Gamma(s+1)Li_{s+1}(e^{\mu}) \end{equation*}

The polylog can  be easily related to the Riemann zeta function

    \[\boxed{Li_s(1)=\zeta(s)}\]

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

Remark: Li_4\left(\dfrac{1}{2}\right) 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)   \begin{equation*} \dfrac{d}{dx}\left(Li_n(x)\right)=\dfrac{1}{x}Li_{n-1}(x) \end{equation*}

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)   \begin{equation*} \begin{split} \dfrac{Li_m(1/64)}{6^{m-1}}-\dfrac{Li_m(1/8)}{3^{m-1}}-\dfrac{2Li_m(1/4)}{2^{m-1}}+\dfrac{4Li_m(1/2)}{9}-\\ -\dfrac{5(-\ln 2)^m}{9m!}+\dfrac{\pi^2(-\ln 2)^{m-2}}{54(m-2)!}-\dfrac{\pi^4(-\ln 2)^{m-4}}{486(m-4)!}-\\ -\dfrac{403\zeta (5)(-\ln 2)^{m-5}}{1296(m-5)!}=0 \end{split} \end{equation*}

 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 n=-1,-2,-3,-4,-5,\ldots,-\infty

In fact, we have

    \[\boxed{Li_{-n}(z)=\sum_{k=1}^\infty k^nz^k=\dfrac{1}{(1-z)^{n+1}}\sum_{i=0}^n\langle\begin{matrix}n\\ i\end{matrix}\rangle z^{n-i}}\]

and where

    \[\langle\begin{matrix}n\\ i\end{matrix}\rangle\]

is an eulerian number.

Polylogs also arise in the theory of generalized harmonic numbers H_{n,q}

    \[\sum_{n=1}^\infty H_{n,q}z^n=\dfrac{Li_q(z)}{1-z}\]

with \vert z\vert<1. There are some special polylogs with special (perhaps subjective) beauty

    \[Li_{-2}(x)=\dfrac{x(x+1)}{(1-x)^3}\]

    \[Li_{-1}(x)=\dfrac{x}{(1-x)^2}\]

    \[Li_{0}(x)=\dfrac{x}{1-x}\]

Note that these are rational functions! We also have

    \[Li_1(x)=-\ln (1-x)\]

    \[Li_n(-1)=-\eta(n)\]

    \[Li_x(1)=\zeta(x)\]

    \[Li_n(1)=\zeta(n)\]

Some interesting known values of the polylog

    \[Li_1\left(\dfrac{1}{2}\right)=\ln (2)\]

    \[Li_2\left(\dfrac{1}{2}\right)=\dfrac{1}{12}\left(\pi^2-6(\ln 2)^2\right)\]

    \[Li_3\left(\dfrac{1}{2}\right)=\dfrac{1}{24}\left[4(\ln 2)^3-2\pi^2\ln 2+21\zeta (3)\right]\]

No higher formulas are known for Li_n(1/2) if n=4,5,\ldots

Euler’s dilogarithm is defined to be

    \[Li_2(z)=\sum_{k=1}^\infty\dfrac{z^k}{k^2}=-\int_0^z\dfrac{\ln (1-t)dt}{t}\]

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

    \[Li_3(z)=\sum_{k=1}^\infty\dfrac{z^k}{k^3}\]

and where

    \[Li_3(-1)=-\dfrac{3}{4}\zeta (3)\]

and the beautiful result

(6)   \begin{equation*} \begin{split} Li_3(z)+Li_3(1-z)+Li_3(1-z^{-1})=\\ \zeta (3)+\dfrac{\ln^3(z)}{6}+\dfrac{\pi^2}{6}\ln(z)-\dfrac{1}{2}\ln^2 z\ln(1-z) \end{split} \end{equation*}

and thus we obtain another wonderful result

    \[\boxed{\zeta (3)=\dfrac{4}{7}\left[2Li_3(2)-\dfrac{\ln 2}{2}\pi^2\right]=\dfrac{8Li_3(2)-2\pi^2\ln 2}{7}}\]

The multidimensional polylog is defined as

(7)   \begin{equation*} Li_{s_1,\ldots,s_m}(z)=\sum_{n_1>\ldots>n_m>0}\dfrac{z^{n_1}}{n_1^{s_1}\cdots n_m^{s_m}} \end{equation*}

Furthermore, the colored polylog is

(8)   \begin{equation*} Li_{s_1,\ldots ,s_k}(z_1,\ldots,z_k)=\sum_{\substack{s_1,\ldots,s_k\\ n_1>\ldots>n_m>0}}\dfrac{z_1^{n_1}\cdots z_k^{n_k}}{n_1^{s_1}\cdots n_k^{s_k}} \end{equation*}

The Nielsen generalized polylogarithm is

    \[S_{n-1,1}(z)=Li_n(z)\]

such as

    \[S_{n,p}(z)=\dfrac{(-1)^{n+p-1}}{(n-1)!p!}\int_0^1\dfrac{(\ln t)^{n-1}\left[\ln (1-zt)\right]^pdt}{t}\]

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

    \[a_k=\int_1^2\dfrac{\ln^k xdx}{x-1}\]

and

    \[a_p=p!\zeta (p+1)-\dfrac{p\left(\ln 2\right)^{p+1}}{p+1}-p!\sum_{k=0}^{p-1}\dfrac{Li_{p+1-k}(1/2)(\ln 2)^k}{k!}\]

with

a_1=\dfrac{\zeta (2)}{2}=\dfrac{\pi^2}{12}, a_2=\dfrac{\zeta (3)}{4},…

 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

    \[\zeta(s)\equiv \zeta_0 (s)=\sum_{n=1}^\infty n^{-s}\]

    \[\zeta_1(s)\equiv P(s)=\sum_{n=1}^\infty p_n^{-s}=\sum_{n=1}^\infty p(n)^{-s}\]

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

    \[\zeta_k(s)=\sum_{n=1}^\infty \left(P^{(k)}(n)\right)^{-s}=\sum_{n=1}^\infty\left(\underbrace{P(P(\cdots))}_\text{k-times}(n)\right)^{-s}\]

Moreover, the prime zeta function of infinite order should be \zeta_\infty (s)=1. The iterated \Omega-power of the Riemann zeta function is

    \[\underbrace{\zeta(\zeta(\cdots))}_\text{$\Omega$- times}=\zeta^{(\Omega(s))}\equiv \Omega_\zeta(s)\]

We could call it the omega zeta function.

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