The Ladder Operators on 1 + 1-de Sitter Space

Abstract

In this paper, we present the annihilation and creation operators for a moving scalar massive particle on 1 + 1-de Sitter space. This presentation is based on coherent states method and Hall-Mitchell approach about annihilation operator for a system which its phase space is $S_{\mathcal {C}}^{n}$. We show that these operators coincide with the Ladder operators for a quantum particle on circle which was presented by Kowalski-Rembielinski-Papaloucas (both cases have the same phase spaces).

Appendix A

1.1 A.1 A Brief About the Group and Algebra of 1 + 1-de Sitter

1 + 1-de Sitter space is a hyperboloid “M_H” embedded in a three-dimensional Minkowski space:

$$ M_{H}\equiv \Big\{x\in \mathcal{R}^{5} ~|~ x.x=(x^{0})^{2}-(x^{1})^{2}-(x^{2})^{2}=-H^{-2}\Big\}~, $$

(55)

where H is the Hubble constant. The associated symmetric group is SO(1, 2) but we use its covering group i.e. unitary group SU(1, 1) that is represented by [14]:

$$ SU(1,1)=\left \{ g=\left( \begin{array}{cc} \alpha & \beta \\ \overline{\beta}&\overline{\alpha} \end{array}\right)~:~\alpha, \beta \in \mathcal{C},\qquad | \alpha |^{2}-| \beta |^{2}=1 \right \}~. $$

(56)

This group act on matrix $\mathcal {X}$ as follows:

$$ \mathcal{X}^{\prime}=g\mathcal{X}g^{\dag}, $$

(57)

where

$$ \begin{array}{@{}rcl@{}} \mathcal{X}=\left( \begin{array}{cc} x^{0} & x^{1}+ix^{2}\\ x^{1}-ix^{2} & x^{0} \end{array}\right),~~~~ g^{\dag}=\left( \begin{array}{cc} \overline{\alpha} & \beta \\ \overline{\beta} & \alpha \end{array}\right). \end{array} $$

The factorization of group SU(1, 1) is represented by [6] :

$$ \begin{array}{@{}rcl@{}} g &=&jl~, \end{array} $$

(58)

$$ \begin{array}{@{}rcl@{}} j&=&j_{1}~j_{2} =\underbrace{\left( \begin{array}{cc} e^{i\frac{\theta}{2}}&0 \\ 0&e^{-i\frac{\theta}{2}} \end{array}\right)}_{\textrm {``space translation"}} \underbrace{\left( \begin{array}{cc} \cosh \frac{\psi}{2} & \sinh \frac{\psi}{2}\\ \sinh \frac{\psi}{2} & \cosh \frac{\psi}{2} \end{array}\right)}_{\textrm {``time translation"}}~, \end{array} $$

$$ \begin{array}{@{}rcl@{}} l&=&\underbrace{\left( \begin{array}{cc} \cosh \frac{\varphi}{2} & i \sinh \frac{\varphi}{2} \\ -i \sinh \frac{\varphi}{2} &\cosh \frac{\varphi}{2} \end{array}\right)}_{\textrm {``Lorentz transformation"}}~. \end{array} $$

Also, three-parameter group SU(1, 1) induce the following vectors in associated Lie algebra “su(1,1)”:

$$ X_{1}=\frac{1}{2}\left( \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right),\qquad X_{2}=\frac{i}{2}\left( \begin{array}{cc} 0 & 1\\ -1 & 0 \end{array}\right),\qquad X_{3}=\frac{i}{2}\left( \begin{array}{cc} 1 & 0\\ 0 & -1 \end{array}\right).\qquad $$

(59)

Namely, an element of su(1,1) is expressed by:

$$ X=\left( \begin{array}{cc} i y& z\\ \overline{z}& -i y \end{array}\right), $$

(60)

where $y \in \mathcal {R}$ and $z \in \mathcal {C}$.

1.2 A.2 Construction of Phase Space by the Orbit Method

In the paper [15], by using the Kirillov orbit method [7,8,9,10], we have shown that the phase space¹ of a scalar massive particle on 1 + 3-de Sitter space is cotangent bundle T^∗(S³) which is isomorphic with the complex sphere “$S_{\mathcal {C}}^{3}$ ”. It is not very difficult to show that the phase space of a scalar massive particle on 1 + 1-de Sitter space is isomorphic with the complex sphere “$S_{\mathcal {C}}^{1}$ ”. For this propose, we choose the point $X_{0}=\left (\begin {array}{cc} 0&1\\ 1&0 \end {array}\right )$ of Lie algebra “su(1,1)” (the case y = 0 and z = 0 of equation (60)). This point is invariant under adjoint action of “time” translation matrix i.e.

$$ \begin{array}{@{}rcl@{}} j_{2}.X_{0}=j_{2}X_{0}j^{-1}_{2}=X_{0}. \end{array} $$

(61)

Therefore, a point of our adjoint orbit is obtained by:

$$ g\cdot X_{0}=lj_{1}~X_{0}j_{1}^{~-1}l^{-1}= \left( \begin{array}{cc} i~\sinh\varphi \cos\theta & \cosh\varphi \cos\theta+i\sin\theta \\ \cosh\varphi \cos\theta-i\sin\theta & -i~\sinh\varphi \cos\theta \end{array}\right)~. $$

(62)

Equations (61) and (62) show that the X₀ is invariant under adjoint action of group SO(1, 1) and therefore the adjoint orbit is identified by:

$$ \mathcal{M}_{H}=SU(1,1)\Big /SO(1,1). $$

(63)

This action (orbit) is transitive and construct a homogeneous space. On the other hand, we know that the homogeneous space for a scalar massive particle on 1 + 1-de Sitter space (on the basis of irreducible representation of group SU(1, 1)) is given by equation (63). Therefore, equation (62) expresses a point of the adjoint orbit for a scalar massive particle on 1 + 1-de Sitter space that corresponds to a point of phase space. By choosing $``~p=m ~\sinh \varphi \cos \limits \theta ~"$, $``~p_{0}=m~\sqrt {\sinh ^{2}\varphi ~\cos \limits ^{2}\theta +1}~"$ and $``~\beta =\frac {1}{2}\arctan \Big (\frac {2 ~\tan \theta ~\cosh \varphi }{\cosh ^{2}\varphi -\tan ^{2}\theta }\Big )~"$ we parameterize (62) as follows:

$$ g\cdot X_{0}:=\left( \begin{array}{cc} iy & z \\ \overline{z} & -iy \end{array}\right)=\left( \begin{array}{cc} i {\displaystyle \frac{p}{m}} & {\displaystyle \frac{p_{0}}{m}}e^{i\beta} \\ {\displaystyle \frac{p_{0}}{m}}e^{-i\beta} & -i {\displaystyle \frac{p}{m}} \end{array}\right)~, $$

(64)

where $p_{0}=\pm \sqrt {m^{2}+p^{2}}$. The quantities β and p play the role position and momentum. This expresses that the adjoint orbit (or phase space) of a scalar massive particle on 1 + 1-de Sitter space is identified with cotangent space T^∗(S¹) that the (β,p) play the role of pair varieties of T^∗(S¹):

$$ \mathrm{T^{*}(S^{1})}=\left \{(\overrightarrow{x},\overrightarrow{p})\in \mathcal{R}^{2} \times \mathcal{R}^{2} |~x^{2}=r^{2},~~\overrightarrow{x}.\overrightarrow{p}=0 \right \}. $$

(65)

On the other hand, from Thiemann complexification method’s (see [1] for $m=r=\hbar =1$) we know that this cotangent space is isomorphic with the complex circle “$S_{\mathcal {C}}^{1}$ ” i.e.

$$ \begin{array}{@{}rcl@{}} S_{\mathcal{C}}^{1}&=&\left \{ \overrightarrow{a}=\cosh(p)\overrightarrow{x}+i~ \frac{\sinh (p)}{p} \overrightarrow{p}\in \mathcal{C}^{2}~~\Big{|}~~a^{2}=1 \right \} \end{array} $$

$$ \begin{array}{@{}rcl@{}} &=& \left \{ \overrightarrow{a}=\big(a_{1},a_{2} \big)= \Big(\cos(\beta+ip),\sin (\beta+ip) \Big) \right \}, \end{array} $$

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where $p^{2}={p_{1}^{2}}+{p_{2}^{2}}$.

1.3 A.3 Calculation of Component of the Annihilation Operators

Just as is mentioned in Section 3, the operator $\hat {A}_{1}$ is given by:

$$ \begin{array}{@{}rcl@{}} \hat{A}_{1} &=& e^{\frac{-\hat{J^{2}}}{2\epsilon}} ~ \hat{X}_{1} ~ e^{\frac{\hat{J^{2}}}{2\epsilon}}= O_{\cos \beta}+[O_{\cos \beta},O_{\frac{p^{2}}{2\epsilon}}] +\frac{1}{2!}[[O_{\cos\beta},O_{\frac{p^{2}}{2\epsilon}}],O_{\frac{p^{2}}{2\epsilon}}] \end{array} $$

$$ \begin{array}{@{}rcl@{}} &&+ \frac{1}{3!}[[[O_{\cos\beta},O_{\frac{p^{2}}{2\epsilon}}],O_{\frac{p^{2}}{2\epsilon}}], O_{\frac{p^{2}}{2\epsilon}}]+... \end{array} $$

(67)

Now we obtain all of commutation relations in the above equation:

$$ \begin{array}{@{}rcl@{}} [O_{\cos\beta},O_{\frac{p^{2}}{2\epsilon}}]&=& \frac{\epsilon}{2}\frac{e^{-\frac{\epsilon}{4}}}{2}{\sum}_{n}\Big((2n+1)|n><n+1| \end{array} $$

$$ \begin{array}{@{}rcl@{}} &&-(2n-1)|n><n-1| \Big), \end{array} $$

(68)

$$ \begin{array}{@{}rcl@{}} [[O_{\cos\beta},O_{\frac{p^{2}}{2\epsilon}}],O_{\frac{p^{2}}{2\epsilon}}]&=&\left( \frac{\epsilon}{2}\right)^{2} \frac{e^{-\frac{\epsilon}{4}}}{2}{\sum}_{n} \Big((2n+1)^{2}|n><n+1| \end{array} $$

$$ \begin{array}{@{}rcl@{}} &&+(2n-1)^{2}|n><n-1| \Big), \end{array} $$

(69)

$$ \begin{array}{@{}rcl@{}} [[[O_{\cos\beta},O_{\frac{p^{2}}{2\epsilon}}],O_{\frac{p^{2}}{2\epsilon}}],O_{\frac{p^{2}}{2\epsilon}}]&=& \left( \frac{\epsilon}{2}\right)^{3} \frac{e^{-\frac{\epsilon}{4}}}{2}{\sum}_{n} \Big((2n+1)^{3}|n><n+1| \end{array} $$

$$ \begin{array}{@{}rcl@{}} &&-(2n-1)^{3}|n><n-1| \Big). \end{array} $$

(70)

By using equations (68)-(70) in (67), we find that

$$ \begin{array}{@{}rcl@{}} \hat{A}_{1} &=& \frac{e^{-\frac{\epsilon}{4}}}{2}{\sum}_{n}\Big(1+\frac{\epsilon}{2}(2n+1) +\frac{1}{2!}\left( \frac{\epsilon}{2}\right)^{2}(2n+1)^{2} \end{array} $$

$$ \begin{array}{@{}rcl@{}} &&+\frac{1}{3!}\left( \frac{\epsilon}{2}\right)^{3}(2n+1)^{3}+....\Big)|n><n+1|+ \frac{e^{-\frac{\epsilon}{4}}}{2}{\sum}_{n}\Big(1-\frac{\epsilon}{2}(2n-1) \end{array} $$

$$ \begin{array}{@{}rcl@{}} &&+\frac{1}{2!}(\frac{\epsilon}{2}\boldsymbol)^{2}(2n-1)^{2}- \frac{1}{3!}\left( \frac{\epsilon}{2}\right)^{3}(2n-1)^{3}+....\Big)|n><n-1| \end{array} $$

$$ \begin{array}{@{}rcl@{}} &=&\frac{e^{-\frac{\epsilon}{4}}}{2}{\sum}_{n}\Big(e^{\frac{\epsilon}{2}(2n+1}\Big)|n><n+1| + \frac{e^{-\frac{\epsilon}{4}}}{2}{\sum}_{n}\Big(e^{-\frac{\epsilon}{2}(2n-1}\Big)|n><n-1| \end{array} $$

$$ \begin{array}{@{}rcl@{}} &=& \frac{e^{\frac{\epsilon}{4}}}{2}{\sum}_{n}\Big(e^{\epsilon n}|n><n+1|+e^{-\epsilon n}|n><n-1| \Big). \end{array} $$

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