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).
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Notes
For simple and semi-simple groups, the adjoint orbit is equivalent to phase space.
References
Hall, B., Mitchell, J.J.: Coherent states on spheres. J. Math. Phys. 43(3), 1211–1236 (2002)
Thiemann, T.: Reality conditions inducing transforms for quantum gauge field theory and quantum gravity. Class Quantum Grav. 13, 1383–1403 (1996)
Thiemann, T.: Gauge field theory coherent states (GCS): I. General properties. Class Quantum Grav. 18, 2025–2064 (2001)
Berezin, F.A.: Quantization. Math.USSR Izvestija 8, 1109–1165 (1974)
Kowalski, K., Rembielinski, J., Papaloucas, L. C.: Coherent states for a quantum particle on a circle. J. Phys. A: Math. Gen. 29, 4149 (1996)
Rabeie, A.: These de doctorat de l’universite de MARNE-LA-VALLEE Physique quantique des systeme elementaires dans de Sitter (2005)
Kirillov, A.A.: Unitary representations of nilpotent lie groups. RUSS MATH SURV 17, 53–104 (1962)
Kirillov, A. A.: Merits and demerits of the orbit method. Bull. Amer. Math. Soc. 36, 433–488 (1999)
Kirillov, A.A.: Elements of the Theory of Representations. Springer, Berlin Heidelberg New York (1976)
Kirillov, A.A.: Lectures on the Orbit Method. American Mathematical Society (2004)
Gazeau, J. P., Lachieze-Rey, M., Piechocki, W.: On three quantization methods for particle on hyperboloid. arXiv:gr-qc/0503060 (2005)
Perelomov, A. M.: Generalized Coherent States and their Applications. Springer, Berlin (1986)
Sugiura, M.: Unitary Representations and Harmonic Analysis. Kodansha Scientific Books (1990)
Vilenkin, N.J.: Special functions and the theory of group representations. American Mathematical Society Providence (1968)
Rabeie, A.: Quantum physics of an elementary system in de Sitter space. Eur. Phys.J.C 72, 2135 (2012)
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Appendix A
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 “MH” embedded in a three-dimensional Minkowski space:
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]:
This group act on matrix \(\mathcal {X}\) as follows:
where
The factorization of group SU(1, 1) is represented by [6] :
Also, three-parameter group SU(1, 1) induce the following vectors in associated Lie algebra “su(1,1)”:
Namely, an element of su(1,1) is expressed by:
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 spaceFootnote 1 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.
Therefore, a point of our adjoint orbit is obtained by:
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:
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:
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):
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.
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:
Now we obtain all of commutation relations in the above equation:
By using equations (68)-(70) in (67), we find that
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Rabeie, A., Rezaei, S. The Ladder Operators on 1 + 1-de Sitter Space. Int J Theor Phys 60, 3850–3860 (2021). https://doi.org/10.1007/s10773-021-04929-3
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DOI: https://doi.org/10.1007/s10773-021-04929-3