Krein space quantization in curved and flat spacetimes

Abstract

We reexamine in detail a canonical quantization methodà la Gupta-Bleuler in which the Fock space is built over a so-called Krein space. This method has already been successfully applied to the massless minimally coupled scalar field in de Sitter spacetime for which it preserves covariance. Here, it is formulated in a more generalcontext. An interesting feature of the theory is that, although the field is obtained by canonical quantization, it is independent of Bogoliubov transformations. Moreover no infinite term appears in the computation of T µν mean values and the vacuum energy of the free field vanishes: 0 | T 00 | 0 = 0. We also investigate the behaviour of the Krein quantization in Minkowski space for a theory with interaction. We show that one can recover the usual theory with the exception that the vacuum energy of the free theory is zero.

A small non-vanishing cosmological constant from the Krein–Gupta–Bleuler vacuum

“…Our aim in the present work, instead, will be to point out a potential relevance between the observable smallness of the cosmological constant and a choice of vacuum in the dS gravitational background of * bamba@sss.fukushima-u.ac.jp † Enayati@iauctb.ac.ir ‡ s.rahbardehghan@iauctb.ac.ir our expanding Universe, now known as the KGB vacuum. This vacuum, based on a new representation of the canonical commutation relations, was recently proposed as an alternative to the dS natural vacuum state (the Bunch-Davies state) that yields a fully covariant and coordinate-independent quantization (the KGB quantization) of linearized gravity in dS space [6][7][8][9][10][11][12]. [Due to the lack of the natural dS-invariant vacuum state for free gravitons, the fact that is now widely accepted in the physics community (see, for instance, [7,13,14]), the usual canonical quantization seems to break down for field theory of dS quantum gravity.]…”

“…Applying the same procedure on other canonically quantizable theories, by restricting V to the positive energy modes denoted by V ′ , would simply imply the fact that the so-called radiative corrections in the KGB context are the same as in the conventional one. In this sense, it seems that vacuum effects in the KGB quantization scheme only involve the interacting vacuum (see the details in [10]).…”

Massless spin-2 field in de Sitter space

“…Here, it must be underlined that, when interaction is present, with respect to the procedure given in [53], determining the space of physical states is the critical step in defining the unitary condition of the theory. Applying the unitary condition, it is proved that this quantization scheme in Minkowski space when interaction is taken into account truly yields the common results; the so-called radiative corrections are indeed the same as usual QFT (see the mathematical details in [53]). It also allows us to obtain the exact usual result for the black hole radiation, even regarding that the free field vacuum expectation value of the energy-momentum tensor is zero [54] (in this regard, see also [55][56][57]).…”

Massive Rarita-Schwinger field in de Sitter space

“…All of these developments plead in favour of setting up a model of QFT in dS spacetime with the same level of completeness and rigor as its Minkowskian counterpart. In this regard, we refer in particular to a promising formulation of such a theory and its subsequent thermic interpretation that was originally put forward for the "massive" scalar fields in dS spacetime in the 1990's [7][8][9], and during recent two decades, it has been subject to scrutiny in a number of works to make explicit the extra algebraic structure inherent to other dS elementary systems (see, for instance, [10][11][12][13][14][15][16][17][18][19][20][21]). Technically, this model of dS QFT enjoys a robust group theoretical content.…”