Supersymmetric Dynamics and Zeta-Functions

Abstract

Boson, fermion, and super oscillators and (statistical) mechanism of cosmological constant; finite approximation of the zeta-function and fermion factorization of the bosonic statistical sum concidered.

Supermathematics unifies discrete and continual aspects of mathematics. Boson oscillator hamiltonian is

$${{H}_{b}} = \hbar \omega {{({{b}^{ + }}b + b{{b}^{ + }})} \mathord{\left/ {\vphantom {{({{b}^{ + }}b + b{{b}^{ + }})} 2}} \right. \kern-0em} 2} = \hbar \omega ({{b}^{ + }}b + a),\,\,\,\,a = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}.$$

((1))

corresponding energy spectrum ${{E}_{{bn}}}$ and eigenfunctions $\left| {{{n}_{b}}} \right\rangle$ are

$$\begin{gathered} {{H}_{b}}\left| {{{n}_{b}}} \right\rangle = {{E}_{{bn}}}\left| {{{n}_{b}}} \right\rangle ,\,\,\,\,{{E}_{{bn}}} = \hbar \omega ({{n}_{b}} + a), \\ {{n}_{b}} = 0,1,2,.... \\ \end{gathered} $$

((2))

Fermion oscillator hamiltonian, eigenvectors and energies are

$$\begin{gathered} {{H}_{f}} = \hbar \omega ({{f}^{ + }}f - {{f{{f}^{ + }})} \mathord{\left/ {\vphantom {{f{{f}^{ + }})} 2}} \right. \kern-0em} 2} = \hbar \omega ({{f}^{ + }}f - a), \\ {{H}_{f}} = \left| {{{n}_{f}}} \right\rangle = {{E}_{{fn}}}\left| {{{n}_{f}}} \right\rangle , \\ {{E}_{{fn}}} = \hbar \omega ({{n}_{f}} - a),\,\,\,\,{{n}_{f}} = 0,1. \\ \end{gathered} $$

((3))

For supersymmetric oscillator we have

$$\begin{gathered} H = {{H}_{b}} + {{H}_{f}},\,\,\,\,H\left| {{{n}_{b}},{{n}_{f}}} \right\rangle = \hbar \omega ({{n}_{b}} + {{n}_{f}})\left| {{{n}_{b}},{{n}_{f}}} \right\rangle , \\ \left| {{{n}_{b}},{{n}_{f}}} \right\rangle = \left| {{{n}_{b}}} \right\rangle \left| {{{n}_{f}}} \right\rangle ,\,\,\,\,{{E}_{{{{n}_{b}},{{n}_{f}}}}} = \hbar \omega ({{n}_{b}} + {{n}_{f}}). \\ \end{gathered} $$

((4))

For background-vacuum $\left| {0,0} \right\rangle ,$ energy ${{E}_{{0,0}}} = 0.$ For higher energy states $\left| {n - 1,1} \right\rangle ,$$\left| {n,0} \right\rangle ,$${{E}_{{n - 1,1}}} = {{E}_{{n,0}}}.$ Supersymmetry needs not only the same frequency for boson and fermion oscillators, but also that $2a = 1.$

A minimal realization of the algebra of supersymmetry

$$\{ Q,{{Q}^{ + }}\} = H,\{ Q,Q\} = \{ {{Q}^{ + }},{{Q}^{ + }}\} = 0,$$

((5))

is given by a point particle dynamics in one dimension, [1]

$$\begin{gathered} Q = f( - iP + {{W)} \mathord{\left/ {\vphantom {{W)} {\sqrt 2 }}} \right. \kern-0em} {\sqrt 2 }},\,\,\,\,{{Q}^{ + }} = {{f}^{ + }}(iP + {{W)} \mathord{\left/ {\vphantom {{W)} {\sqrt 2 }}} \right. \kern-0em} {\sqrt 2 }}, \\ P = {{ - i\partial } \mathord{\left/ {\vphantom {{ - i\partial } {\partial x}}} \right. \kern-0em} {\partial x}}, \\ \end{gathered} $$

((6))

where the superpotential $W(x)$ is any function of x, and spinor operators f and ${{f}^{ + }}$ obey the anticommuting relations

$$\{ f,{{f}^{ + }}\} = 1,\,\,\,\,{{f}^{2}} = {{({{f}^{ + }})}^{2}} = 0.$$

((7))

There is a following representation of operators f, ${{f}^{ + }}$ and σ by Pauli spin matrices

$$\begin{gathered} f = \frac{{{{\sigma }_{1}} - i{{\sigma }_{2}}}}{2} = \left( {\begin{array}{*{20}{c}} 0&0 \\ 1&0 \end{array}} \right),\,\,\,\,{{f}^{ + }} = \frac{{{{\sigma }_{1}} + i{{\sigma }_{2}}}}{2} = \left( {\begin{array}{*{20}{c}} 0&1 \\ 0&0 \end{array}} \right), \\ \sigma = {{\sigma }_{3}} = \left( {\begin{array}{*{20}{c}} 1&0 \\ 0&{ - 1} \end{array}} \right). \\ \end{gathered} $$

((8))

From formulae (5) and (6) then we have

$$H = ({{P}^{2}} + {{W}^{2}} + {{\sigma {{W}_{x}})} \mathord{\left/ {\vphantom {{\sigma {{W}_{x}})} 2}} \right. \kern-0em} 2}.$$

((9))

The simplest nontrivial case of the superpotential $W = \omega x$ corresponds to the supersimmetric oscillator with Hamiltonian

$$\begin{gathered} H = {{H}_{b}} + {{H}_{f}},\,\,\,\,{{H}_{b}} = ({{P}^{2}} + {{\omega }^{2}}{{{{x}^{2}})} \mathord{\left/ {\vphantom {{{{x}^{2}})} 2}} \right. \kern-0em} 2}, \\ {{H}_{f}} = {{\omega \sigma } \mathord{\left/ {\vphantom {{\omega \sigma } 2}} \right. \kern-0em} 2}. \\ \end{gathered} $$

((10))

The ground state energies of the bosonic and fermionic parts are

$${{E}_{{b0}}} = {\omega \mathord{\left/ {\vphantom {\omega 2}} \right. \kern-0em} 2},\,\,\,\,{{E}_{{f0}}} = {{ - \omega } \mathord{\left/ {\vphantom {{ - \omega } 2}} \right. \kern-0em} 2},$$

((11))

so the vacuum energy of the supersymmetric oscillator is

$$\begin{gathered} \left\langle 0 \right|H\left| 0 \right\rangle = {{E}_{0}} = {{E}_{{b0}}} + {{E}_{{f0}}} = 0, \\ \left| 0 \right\rangle = \left| {{{n}_{b}},{{n}_{f}}} \right\rangle = \left| {{{n}_{b}}} \right\rangle \left| {{{n}_{f}}} \right\rangle . \\ \end{gathered} $$

((12))

Let us see on this toy—solution of the cosmological constant problem from the quantum statistical viewpoint. The statistical sum of the supersymmetric oscillator is

$$Z(\beta ) = {{Z}_{b}}{{Z}_{f}},$$

((13))

where

$$\begin{gathered} {{Z}_{b}} = \sum\limits_n {{{e}^{{ - \beta {{E}_{{bn}}}}}}} = {{e}^{{{{ - \beta \omega } \mathord{\left/ {\vphantom {{ - \beta \omega } 2}} \right. \kern-0em} 2}}}} + {{e}^{{ - \beta \omega (1 + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2})}}} \\ + \,\,... = {{{{e}^{{{{ - \beta \omega } \mathord{\left/ {\vphantom {{ - \beta \omega } 2}} \right. \kern-0em} 2}}}}} \mathord{\left/ {\vphantom {{{{e}^{{{{ - \beta \omega } \mathord{\left/ {\vphantom {{ - \beta \omega } 2}} \right. \kern-0em} 2}}}}} {(1 - {{e}^{{ - \beta \omega }}})}}} \right. \kern-0em} {(1 - {{e}^{{ - \beta \omega }}})}}, \\ {{Z}_{f}} = \sum\limits_n {{{e}^{{ - \beta {{E}_{{fn}}}}}}} = {{e}^{{{{\beta \omega } \mathord{\left/ {\vphantom {{\beta \omega } 2}} \right. \kern-0em} 2}}}} + {{e}^{{{{ - \beta \omega } \mathord{\left/ {\vphantom {{ - \beta \omega } 2}} \right. \kern-0em} 2}}}}. \\ \end{gathered} $$

((14))

In the low temperature limit,

$$Z(\beta ) = 1 + O({{e}^{{ - \beta \omega }}}) \to 1,\,\,\,\,\beta = {{T}^{{ - 1}}},$$

((15))

so cosmological constant $\lambda \sim \ln Z \to 0.$ From observable values of β and the cosmological constant we estimate ω.

The Riemann zeta function (RZF) can be interpreted in thermodynamic terms as a statistical sum of a system with energy spectrum: ${{E}_{n}} = \ln n,\,\,\,\,n = 1,2,...\,:$

$$\begin{gathered} \zeta (s) = \sum\limits_{n \geqslant 1} {{{n}^{{ - s}}}} = Z(\beta ) = \sum\limits_{n \geqslant 1} {\exp ( - \beta {{E}_{n}})} , \\ \beta = s,\,\,\,\,{{E}_{n}} = \ln n,\,\,\,\,n = 1,2,.... \\ \end{gathered} $$

((16))

Let us consider the following finite approximation of RZF

$$\begin{gathered} {{\zeta }_{N}}(s) = \sum\limits_{n = 1}^N {{{n}^{{ - s}}}} = \frac{1}{{\Gamma (s)}}\int\limits_0^\infty {dt{{t}^{{s - 1}}}} \frac{{{{e}^{{ - t}}} - {{e}^{{ - (N + 1)t}}}}}{{1 - {{e}^{{ - t}}}}} \\ = \zeta (s) - {{\Delta }_{N}}(s),\,\,\,\,\operatorname{Re} s > 1, \\ \zeta (s) = \frac{1}{{\Gamma (s)}}\int\limits_0^\infty {dt} \frac{{{{t}^{{s - 1}}}}}{{{{e}^{t}} - 1}},\,\,\,\,{{\Delta }_{N}}(s) = \frac{1}{{\Gamma (s)}}\int\limits_0^\infty {dt} \frac{{{{t}^{{s - 1}}}{{e}^{{ - Nt}}}}}{{{{e}^{t}} - 1}}. \\ \end{gathered} $$

((17))

Another formula, which can be used on critical line, is

$$\begin{gathered} \zeta (s) = {{(1 - {{2}^{{1 - s}}})}^{{ - 1}}}\sum\limits_{n \geqslant 1} {{{{( - 1)}}^{{n + 1}}}} {{n}^{{ - s}}} \\ = \frac{1}{{1 - {{2}^{{1 - s}}}}}\frac{1}{{\Gamma (s)}}\int\limits_0^\infty {\frac{{{{t}^{{s - 1}}}dt}}{{{{e}^{t}} + 1}}} ,\,\,\,\,\operatorname{Re} s > 0. \\ \end{gathered} $$

((18))

Corresponding finite approximation of RZF is

$$\begin{gathered} {{\zeta }_{N}}(s) = {{(1 - {{2}^{{1 - s}}})}^{{ - 1}}}\sum\limits_{n = 1}^N {{{{( - 1)}}^{{n - 1}}}{{n}^{{ - s}}}} = \frac{1}{{1 - {{2}^{{1 - s}}}}}\frac{1}{{\Gamma (s)}} \\ \times \,\,\int\limits_0^\infty {\frac{{{{t}^{{s - 1}}}(1 - {{{( - {{e}^{{ - t}}})}}^{N}})dt}}{{{{e}^{t}} + 1}}} = \zeta (s) - {{\Delta }_{N}}(s), \\ {{\Delta }_{N}}(s) = \frac{1}{{\Gamma (s)}}\int\limits_0^\infty {dt\frac{{{{t}^{{s - 1}}}{{{( - {{e}^{{ - t}}})}}^{N}})}}{{{{e}^{t}} + 1}}} \sim \pm {{N}^{{ - s}}} \\ \end{gathered} $$

((19))

at a (nontrivial) zero of RZF, ${{s}_{0}},$${{\zeta }_{N}}({{s}_{0}}) = - {{\Delta }_{N}}({{s}_{0}}).$ In the integral form, dependence on N is analytic and we can consider any complex valued N. It is interesting to see dependence (evolution) of zeros with N. For the simplest nontrivial integer $N = 2,$

$$\begin{gathered} {{\zeta }_{2}}(s) = {{(1 - {{2}^{{1 - s}}})}^{{ - 1}}}(1 - {{2}^{{ - s}}}) \\ = \frac{{1 - {{2}^{{ - s}}}}}{{1 - {{2}^{{1 - s}}}}} = \frac{{{{2}^{s}} - 1}}{{{{2}^{s}} - 2}} = \frac{{{{2}^{{s - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} - {1 \mathord{\left/ {\vphantom {1 {\sqrt 2 }}} \right. \kern-0em} {\sqrt 2 }}}}{{{{2}^{{s - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} - \sqrt 2 }}, \\ \end{gathered} $$

((20))

we have zeros at $s = {{2\pi in} \mathord{\left/ {\vphantom {{2\pi in} {\ln 2}}} \right. \kern-0em} {\ln 2}},\,\,\,\,\,n = 0, \pm 1, \pm 2,....$

Let as consider the following formula (Qvelementar particles)

$$\frac{1}{{1 - q}} = (1 + q)(1 + {{q}^{2}})(1 + {{q}^{4}})...,\,\,\,\,\left| q \right| < 1,$$

((21))

which can be proved as

$$\begin{gathered} {{p}_{k}} \equiv (1 + q)(1 + {{q}^{2}})(1 + {{q}^{4}}) \ldots (1 + {{q}^{{{{2}^{k}}}}}) \\ = \frac{{1 - {{q}^{{{{2}^{{(k + 1)}}}}}}}}{{1 - q}},\,\,\,\,c(1 - {{\left| q \right|}^{{{{2}^{{(k + 1)}}}}}}) < \left| {{{p}_{k}}} \right| < c(1 + {{\left| q \right|}^{{{{2}^{{(k + 1)}}}}}}), \\ \mathop {lim}\limits_{k \to \infty } \left| {{{p}_{k}}} \right| = c = {1 \mathord{\left/ {\vphantom {1 {\left| {1 - q|} \right|}}} \right. \kern-0em} {\left| {1 - q|} \right|}},\,\,\,\,\mathop {lim}\limits_{k \to \infty } {{p}_{k}} = {1 \mathord{\left/ {\vphantom {1 {(1 - q)}}} \right. \kern-0em} {(1 - q)}}. \\ \end{gathered} $$

((22))

The formula (21) reminds us the boson and fermion statsums

$$\begin{gathered} {{Z}_{b}} = \frac{{{{q}^{a}}}}{{1 - q}},\,\,\,\,{{Z}_{f}} = \frac{{1 + q}}{{{{q}^{a}}}},\,\,\,\,q = \exp ( - \beta \hbar \omega ), \\ a = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2},\,\,\,\,\beta = {1 \mathord{\left/ {\vphantom {1 T}} \right. \kern-0em} T}, \\ \end{gathered} $$

((23))

and can be transformed in the following relation

$${{Z}_{b}}(\omega ) = {{Z}_{f}}(\omega ){{Z}_{f}}(2\omega ){{Z}_{f}}(4\omega )....$$

((24))

Indeed,

$$\begin{gathered} {{Z}_{b}}(\omega ) = \frac{{{{q}^{a}}}}{{1 - q}} = {{q}^{b}}{{Z}_{f}}(\omega ){{Z}_{f}}(2\omega ){{Z}_{f}}(4\omega )..., \\ b = 2a + 2a(1 + 2 + {{2}^{2}} + ...) \\ = 2a\left( {1 + \frac{1}{{1 - 2}}} \right) = 0,\,\,\,\,\,{{\left| 2 \right|}_{2}} = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}, \\ \end{gathered} $$

((25))

where ${{\left| n \right|}_{p}} = {1 \mathord{\left/ {\vphantom {1 {{{p}^{k}}}}} \right. \kern-0em} {{{p}^{k}}}},\,\,\,\,n = {{p}^{k}}m,$ is p-adic norm of $n,k$ is the number of p-prime factors of n.

Bytheway we have an extra bonus! We see that the fermion content of the boson wears the p-adic sense [2]. The prime $p = 2,$ in this case. Also, the vacuum energy of the oscillators wear p-adic sense.