The issue of branched Hamiltonian in F(T) teleparallel gravity

Abstract

As in the case of Lanczos–Lovelock gravity, the main advantage of [Formula: see text] gravity is said to be that it leads to second-order field equations, while [Formula: see text] gravity theory leads to fourth-order equations. We show that it is rather a disadvantage, since it leads to the unresolved issue of ‘Branched Hamiltonian’. The problem is bypassed in [Formula: see text] gravity theory.

Perusing Buchbinder–Lyakhovich Canonical Formalism for Higher-Order Theories of Gravity

“…We quote from [22] the general argument in connection with the total derivative terms: "it is just the change of the variables in the wave function and the phase transformation, plus the change of the integration measure, and the transformation of the momenta respecting the change of the measure, and so a unitary transformation relates the two". It is possible (although we have not found this) that each pair of quantum equations cast from ( 27) and ( 28), ( 31) and (32), and ( 45) and ( 47) are related by unitary transformation. However, it was also mentioned [22] that different forms of coupling parameter yield different quantum dynamics in the case of MHF, due to the presence of a coupling term ( f (φ)p φ ) for the non-minimally coupled case, and (γ (φ)p φ ) for the Einstein-Gauss-Bonnet-dilaton coupled case, in the Hamiltonian.…”

“…Nevertheless, here the difference is predominant and explicit. Note that the f (φ) term does not appear in (31), while it is coupled with p φ in (32). This coupled ( f (φ)p φ ) term requires operator ordering in the quantum domain, which is different for different forms of f (φ).…”

“…The same is true for 'dilatonic coupled Gauss-Bonnet-theory in the presence of higher-order terms', where additional classical correspondence from the quantum counterpart cannot be established [22]. In view of such a situation, yet another technique was developed under the name 'modified Horowitz formalism' (MHF), which was successfully applied in different modified higher-order theories of gravity, to explore the evolution of the very early universe [15,[17][18][19][22][23][24][25][26][27][28][29][30][31][32]. In MHF, the action is expressed in terms of the three space curvature (instead of the scale factor), 'the total derivative terms' are removed by integrating the action by parts, and Horowitz's formalism (introduction of auxiliary variable and else) is followed, thereafter.…”