Quantum cosmology: From hidden symmetries towards a new (supersymmetric) perspective

Abstract

We review pedagogically some of the basic essentials regarding recent results intertwining boundary conditions, the algebra of constraints and hidden symmetries in quantum cosmology. They were extensively published in Refs. [S. Jalalzadeh, S. M. M. Rasouli and P. V. Moniz, Phys. Rev. D 90 (2014) 023541, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 89 (2014), S. Jalalzadeh, T. Rostami and P. V. Moniz, Eur. Phys. J. C 75 (2015) 38, arXiv:gr-qc/1412.6439 and T. Rostami, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 92 (2015) 023526, arXiv:gr-qc/1507.04212], where complete discussions and full details can be found. More concretely, in Refs. [S. Jalalzadeh, S. M. M. Rasouli and P. V. Moniz, Phys. Rev. D 90 (2014) 023541, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 89 (2014) and S. Jalalzadeh, T. Rostami and P. V. Moniz, Eur. Phys. J. C 75 (2015) 38, arXiv:gr-qc/1412.6439] it has been shown that specific boundary conditions can be related to the algebra of Dirac observables. Moreover, a process afterwards associated to the algebra of existent hidden symmetries, from which the boundary conditions can be selected, was introduced. On the other hand, in Ref. [T. Rostami, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 92 (2015) 023526, arXiv:gr-qc/1507.04212] it was subsequently argued that some factor ordering choices can be extracted from the hidden symmetries structure of the minisuperspace model. In Refs. [S. Jalalzadeh, S. M. M. Rasouli and P. V. Moniz, Phys. Rev. D 90 (2014) 023541, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 89 (2014), S. Jalalzadeh, T. Rostami and P. V. Moniz, Eur. Phys. J. C 75 (2015) 38, arXiv:gr-qc/1412.6439 and T. Rostami, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 92 (2015) 023526, arXiv:gr-qc/1507.04212], we proceeded gradually towards less simple models, ranging from a FLRW model with a perfect fluid [S. Jalalzadeh, S. M. M. Rasouli and P. V. Moniz, Phys. Rev. D 90 (2014) 023541] up to a conformal scalar field content [T. Rostami, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 92 (2015) 023526, arXiv:gr-qc/1507.04212]. We envisage that we could extend this framework towards a class of shape invariant potentials, which could include well known analytically solvable cosmological cases. Provided, we identify integrability in terms of the shape invariance conditions, we could eventually consider to import features of supersymmetric quantum mechanics towards quantum cosmology [P. V. Moniz, Quantum Cosmology-the Supersymmetric Perspective-Vol. 1: Fundamentals, Lecture Notes in Physics, Vol. 803 (Springer-Verlag, Berlin, 2010), P. V. Moniz, Quantum Cosmology-the Supersymmetric Perspective-Vol. 2: Advanced Topics, Lecture Notes in Physics, Vol. 804 (Springer, New York, 2010)], which we will also discuss in this review. Another point to emphasize is that by means of a hidden symmetry and then an algebra of Dirac observables, boundary conditions are extracted (and not ad hoc formulated) within a framework intrinsic to each model dynamics. Therefore, meeting DeWitt’s conjecture [B. S. DeWitt, Phys. Rev. 160 (1967) 1113] that “the constraints are everything” and nothing else but the constraints should be needed.

Quantum cosmology: The (supersymmetric) door

“…Note that the above relations are known as the commutators of the Lie algebra of su (1,1). In this representation, this Lie algebra is labeled by a positive discrete …”

“…Nevertheless, other boundary conditions can be put forward with well known results, which have been widely investigated in the literature, including consistency and potential observational features. It would be interesting to find some algebraic support for them, along the lines discussed in this review (and [1][2][3][4][5] ), but the statements defining the main boundary conditions are more of a 'topological' nature for the minisuperspaces involved and hence it is not obvious whether this can be achieved. Moreover, singularities of a differential nature can be present in a given cosmology and a discussion involving them as well as conditions on Ψ and admissible boundary conditions by means of the algebra of constraints is indeed quite worthy to venture into, namely within the context of hidden symmetries and 'beyond' (e.g., supersymmetry).…”

“…The su (1,1) hidden symmetry of that model with the set of gauge invariant of the Bargmann values { In Ref. 4 , by a similar procedure, the hidden symmetries present in a pre-big bang model were identified, namely u(1, 1) together with time reversal and parity.…”

“…We further showed that the Bargmann indices are gauge invariant observables of the quantum cosmological model. From the vanishing of the commutator of the su (1,1) generator with the Hamiltonian of the system, in addition to the gauge invariance of the Bargmann indices, it has been possible to select the wave function of the Universe. In other words, our proposed framework (cf.…”

Classical universe emerging from quantum cosmology without horizon and flatness problems

“…To obtain a Bohmian interpretation for these models, the usual procedure is to construct a wave packet by superposition of eigenstates [33][34][35][36][37][38]. But it is not clear that the hidden symmetries of a model give us permission in general to construct such wave packets [39][40][41][42]. On the other hand, the CQHJ interpretation gives us an opportunity to obtain a causal interpretation even for real wave functions of the universe.…”