Quantum Gowdy T ³ model: a uniqueness result

Abstract

Modulo a homogeneous degree of freedom and a global constraint, the linearly polarised Gowdy T 3 cosmologies are equivalent to a free scalar field propagating in a fixed nonstationary background. Recently, a new field parameterisation was proposed for the metric of the Gowdy spacetimes such that the associated scalar field evolves in a flat background in 1+1 dimensions with the spatial topology of of S 1 -translations. These translations are precisely those generated by the global constraint that remains on the Gowdy model. It is also shown that the proof of uniqueness in the choice of complex structure can be applied to more general field dynamics than that corresponding to the Gowdy cosmologies.

Quantum GowdyT3model: Schrödinger representation with unitary dynamics

“…IV B). Moreover, as we mentioned in the introduction, much stronger results have indeed been proven regarding the uniqueness of the quantization [6,9].…”

“…It is clear that, when expressed in terms of the pairs f b k t f ; b k t f g, the complex structure j t f adopts the same form as j 0 in terms of the pairs f b k ; b k g [namely, it is given by a block-diagonal matrix with the 2 2 blocks j t f k diag i; ÿi ]. Therefore, the j t f representation is such that the classical variables which are quantized as the creation and annihilation operators are f b k t f g and f b k t f g, respectively, rather than fb k g and fb k g. 9 Returning to the covariant description for a moment, the family fj t f g determines a family of complex structures on the covariant phase space via the isomorphism I E 0 (35). These are given by J t f t f ;t 0 J 0 ÿ1 where G t f k t f ;t 0 G t 0 k are the time-evolved modes.…”

“…9 Let us point out that a different but equivalent way to recast time evolution is with the family of representations arising from the set of complex structures fj t f ÿ1 j 0 g. In that case, the annihilation and creationlike variables defined byj t f are fb k t f g and fb k t f g, introduced in Eq. (15).…”

Unique Fock quantization of scalar cosmological perturbations

“…We will now show that any such complex structure gives rise to a representation which is unitarily equivalent to the one defined by J 0 . The necessary and sufficient condition for the unitary implementation of the symplectomorphism U with respect to this new complex structure is that the transformation K −1 UK (obtained with a change of basis as discussed above) be unitarily implementable with respect to the original complex structure J 0 [16]. Of course, the matrix representation of K −1 UK also consists of blocks, which we can write as…”

“…In both cases, the complex structure that would be natural if the field were massless allows for a unitary quantum implementation of the field dynamics and, besides, shares the symmetry of the spatial sections of the auxiliary spacetime (which the vacuum state inherits). In fact, these two properties select a unique unitary equivalence class of Fock representations: all the representations with these attributes are unitarily equivalent [16][17][18]. Furthermore, if we change the field description by performing a linear canonical transformation that scales the field by a time-dependent factor, unitarity is lost: there is no Fock representation for the new canonical pair of field variables in which dynamics is represented in terms of a unitary operator [19].…”