Imprints of microcausality violation on the cosmic microwave background

We consider a modification of the Heisenberg algebra with the non-vanishing commutator of scalar field operators. We then identify the scalar field with the second quantized inflaton fluctuation and calculate effects of microcausality violation on the temperature anisotropy of cosmic microwave background radiation. In the local relativistic invariant quantum field theory (in flat Minkowski space-time) microcausality is encoded in the equal time commutator, [φ(~x, t), φ(~y, t)] = 0 , (1) of a Heisenberg field operator φ(x). The non-vanishing right hand side of (1), besides the violation of microcausality, might cause violation of locality and relativistic invariance. In curved spacetime the situation is more complicated. The background metric breaks relativistic invariance and also the notion of locality is lost. Nevertheless, microcausality can still hold [1]. Heuristically, this can be understood by recalling the basic property of the (pseudo)Riemannian geometry. In the small vicinity of any point of a curved manifold the space-time can be treated as being approximately Minkowskian, and thus the condition (1) applies at small scales. However, one can envisage the situation when the notions of microcausality, locality and relativistic invariance hold only approximately at large distances, and are violated at small distances. Whether the above notions are indeed the basic properties of nature must be ultimately verified by experiments. In principle, miniscule quantum effects can be detected in astronomical observations on temperature 1 anisotropies of the cosmic microwave background (CMB) radiation, providing our universe has undergone a period of rapid inflationary expansion at early stages of its evolution. During the inflationary era large-amplitude, small-scale quantum fluctuations of an inflaton field are stretched over the large macroscopic distances due to the rapid expansion. These quantum fluctuations are imprinted in the CMB. Therefore, one can hope to extract valuable information on quantum dynamics of the inflaton field by studying CMB temperature anisotropies. In this paper we would like to consider non-trivial commutator of fields (and thus violation of microcausality), [φ(~x, η), φ(~y, η)] = i a2(η) μf(~x− ~y) , (2) in expanding conformally flat space-time with the metric, ds = a(η)(dη − d~x) . (3) In (2) f(~x−~y) is an odd function of its argument, f(~x−~y) = −f(~y−~x)1, and μ is some parameter with a dimension dim[φ]−dim[f ]/2. Since f(~x−~y) is the odd function, the commutator (2) does not preserve rotational invariance. Apart from the modified commutator in (3), we also have the standard commutators, [φ(~x, η), π(~y, η)] = i a(η) δ(~x− ~y) , (4) and, [π(~x, η), π(~y, η)] = 0 , (5) where π(x) is the operator of canonical momentum. Now observe that the above pair of operators (φ(x), π(x)) with the deformed Heisenberg algebra (3), (4), (5), can be mapped onto the canonical pair (ψ(x), π(x)) with the ordinary Heisenberg algebra, through the following relation, φ(x) = ψ(x)− μ 2 2 ∫ f(~x− ~ ξ)π(~ ξ, η)d~ ξ . (6) Note that when considering (φ, π) as a canonical pair, the usual Lagrangian gets modified correspondingly. We do not need the exact form of this new Lagrangian here, because all correlators can be expressed through the correlators of the standard canonical pair (ψ, π) through the mapping (6). We also assume that the classical background solution holds in such a new theory. We identify φ(~x, η) with the second-quantized fluctuation of an inflaton field and are interested in computing two-point function 〈0|φ(~x, η)φ(~y, η)|0〉 using the map (6). The power spectrum corresponding to this correlator essentially describes cosmological perturbations which are observed through the CMB temperature fluctuations [2]. Since the field ψ is the standard quantum inflaton field, we have, ψ(x) = ∫