Z_p-Graded Wigner Algebra : Coherent States and Thermodynamics

Abstract

In this paper we extend the ordinary parity into more general one which we call a Z_p-graded parity. Using this we present Z_p-graded Wigner algebra. We discuss the coherent states for Z_p-graded Wigner algebra. Finally we discuss thermodynamics for Z_p-boson.

Appendices

Appendix A

Now let us derive the (43). For j = 0 we have

$$ [ pk ]_p = pk $$

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For j ≠ 0 we have

$$ \begin{array}{@{}rcl@{}} [pk+j ]_p &=& pk +j + \nu \sum\limits_{r=1}^{p-1} \frac{ 1 - q^{-rj}}{1-q^{-r}} \\ &=&pk+j + \nu (p-1) + \nu \sum\limits_{r=1}^{p-1} \sum\limits_{s=1}^{j-1} q^{-rs} \\ &=&pk+j + \nu (p-1) + \nu \sum\limits_{s=1}^{j-1} \sum\limits_{r=1}^{p-1} q^{-rs} \\ &=& pk+j + (p-1) \nu - \nu (j-1) \\ &=& pk +j + (p-j) \nu \end{array} $$

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which completes the derivation of the (43).

Appendix B

Now let us prove the (45). We have for n = pk

$$ \begin{array}{@{}rcl@{}} \sum\limits_{l=0}^{p-1} [ n + l]_p &=&\sum\limits_{l=0}^{p-1} [ pk +j + l]_p \\ &=& pk + \sum\limits_{l=1}^{p-1} (p k + l + (p-l) \nu ) \\ &=& p \left[ pk + \frac{p-1}{2} (1 + \nu) \right] \end{array} $$

We have for n = pk + j (j≠ 0 )

$$ \begin{array}{@{}rcl@{}} \sum\limits_{l=0}^{p-1} [ n + l]_p &=&\sum\limits_{l=0}^{p-1} [ pk +j + l]_p \\ &=& \sum\limits_{l=j}^{p-1} \left( pk + l + (p - l ) \nu \right) + p (k+1) + \sum\limits_{l=1}^{j-1} (p (k+1) + l + (p-l) \nu ) \\ &=& p \left[ pk + j + \frac{p-1}{2} (1 + \nu) \right] \end{array} $$

Appendix C

For k ≥ 1 We have

$$ \begin{array}{@{}rcl@{}} a^p |pk + j \rangle &=& \sqrt{ \prod\limits_{l=0}^{p-1} [pk+j-l]_p} |p(k-1) + j \rangle \\ &=& \sqrt{[pk+j]_p [pk+j-1]_p{\cdots} [pk+1]_p [pk]_p [pk - 1]_p {\cdots} [pk - (p - j - 1)]_p } |p(k - 1) + j \rangle \\ &=& \sqrt{ [pk]_p \prod\limits_{l=1}^{j} [pk+l]_p \prod\limits_{m=1}^{p-j-1} [pk-m]_p } |p(k-1) + j \rangle \\ &=& \sqrt{ pk \prod\limits_{l=1}^{j} (pk + l + (p-l) \nu) \prod\limits_{m=1}^{p-j-1} (pk -m + m \nu) } |p(k-1) + j \rangle \end{array} $$

Appendix D

We have

$$ \begin{array}{@{}rcl@{}} \langle N \rangle_p &=& \langle z | N | z \rangle \\ &=& (e^{(p)} (|z|^2))^{-1} \sum\limits_{j=0}^{p-1} \sum\limits_{k=0}^{\infty} (pk+j) \frac{|z|^{2 (pk+j)}}{[pk+j]_p!} \\ &=& (e^{(p)} (|z|^2))^{-1} \sum\limits_{j=0}^{p-1} \sum\limits_{k=0}^{\infty} pk \frac{|z|^{2 (pk+j)}}{[pk+j]_p!} + (e^{(p)} (|z|^2))^{-1} \sum\limits_{j=0}^{p-1} \sum\limits_{k=0}^{\infty} j \frac{|z|^{2 (pk+j)}}{[pk+j]_p!} \\ &=& (e^{(p)} (|z|^2))^{-1} \sum\limits_{j=0}^{p-1} \sum\limits_{k=0}^{\infty} pk \frac{|z|^{2 (pk+j)}}{[pk+j]_p!} + (e^{(p)} (|z|^2))^{-1} \sum\limits_{j=0}^{p-1} j C_j^{(p)}(|z|^2) \end{array} $$

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Here

$$ \begin{array}{@{}rcl@{}} I &=& \sum\limits_{k=0}^{\infty} pk \frac{|z|^{2 (pk+j)}}{[pk+j]_p!} \\ &=& \sum\limits_{k=0}^{\infty} \frac{pk (|z|^2)^{pk+j}}{ [j]_p! p^{pk} k! {\prod}_{l=1}^j \left( 1 + \frac{l+ (p-l)\nu}{p} \right)_k {\prod}_{m=1}^{p-j-1} \left( 1 + \frac{m \nu -m }{p} \right)_k } \\ &=& \sum\limits_{k=0}^{\infty} \frac{ (|z|^2)^{p(k+1)+j}}{ [j]_p! p^{p(k+1)} k! {\prod}_{l=1}^j \left( 1 + \frac{l+ (p-l)\nu}{p} \right)_{k+1} {\prod}_{m=1}^{p-j-1} \left( 1 + \frac{m \nu -m }{p} \right)_{k+1} } \end{array} $$

Using (a)_k+ 1 = a(a + 1)_k we get

$$ \begin{array}{@{}rcl@{}} I &=& \frac{|z|^{2p}}{ p^{p-1}{\prod}_{l=1}^j \left( 1 + \frac{l+ (p-l)\nu}{p} \right) {\prod}_{m=1}^{p-j-1} \left( 1 + \frac{m \nu -m }{p} \right)} \sum\limits_{k=0}^{\infty} \frac{ (|z|^2)^{pk+j}}{ [j]_p! p^{pk} k! {\prod}_{l=1}^j \left( 2 + \frac{l+ (p-l)\nu}{p} \right)_{k} {\prod}_{m=1}^{p-j-1} \left( 2 + \frac{m \nu -m }{p} \right)_{k} } \\ &=& \frac{|z|^{2p} [p]_p [j]_p!}{ [p+j]_p!} C_j^{(p, 2)} (|z|^2) \end{array} $$

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