Fourier representation for the two-point function of the two-dimensional massless scalar field

1991 ◽  
Vol 22 (3) ◽  
pp. 235-238
Author(s):  
H. G. Embacher ◽  
G. Gr�bl
Keyword(s):  
Scalar Field ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Two Dimensional ◽  
Point Function ◽  
Fourier Representation

scholarly journals The spectral dimension in 2D CDT gravity coupled to scalar fields

2015 ◽  
Vol 30 (13) ◽  
pp. 1550077 ◽  
Author(s):  
J. Ambjørn ◽  
A. Görlich ◽  
J. Jurkiewicz ◽  
H. Zhang
Keyword(s):  
Scalar Field ◽  
Scalar Fields ◽  
Spectral Dimension ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Two Dimensional ◽  
Imaginary Time ◽  
Causal Dynamical Triangulations ◽  
Pure Gravity ◽  
Dynamical Triangulations

Causal Dynamical Triangulations (CDT) provide a non-perturbative formulation of Quantum Gravity assuming the existence of a global time foliation. In our earlier study we analyzed the effect of including d copies of a massless scalar field in the two-dimensional CDT model with imaginary time. For d > 1 we observed the formation of a "blob", somewhat similar to that observed in four-dimensional CDT without matter. In the two-dimensional case the "blob" has a Hausdorff dimension DH = 3. In this paper, we study the spectral dimension DS of the two-dimensional CDT-universe, both for d = 0 (pure gravity) and d = 4. We show that in both cases the spectral dimension is consistent with DS = 2.

scholarly journals MASSLESS SCALAR FIELD PROPAGATOR IN A QUANTIZED SPACETIME

2007 ◽  
Vol 16 (01) ◽  
pp. 51-57 ◽  
Author(s):  
V. ELIAS ◽  
T. G. STEELE ◽  
K. TANAKA
Keyword(s):  
Scalar Field ◽  
Spectral Function ◽  
Imaginary Part ◽  
Single Particle ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Point Function ◽  
Particle Propagation ◽  
Free Fields ◽  
Analytic Behaviour

We consider in detail the analytic behaviour of the non-interacting massless scalar field two-point function in H. S. Snyder's discretized non-commuting spacetime. The propagator we find is purely real on the Euclidean side of the complex p2 plane and goes like 1/p2 as p2→0 from either the Euclidean or Minkowski side. The real part of the propagator goes smoothly to zero as p2 increases to the discretization scale 1/a2and remains zero for p2>1/a2. This behaviour is consistent with the termination of single-particle propagation on the ultraviolet side of the discretization scale. The imaginary part of the propagator, consistent with a multiparticle-state spectral function branch discontinuity, is finite and continuous on the Minkowski side, slowly falling to zero when 1/a2<p2<∞. The multi-particle aspect of this spectral function within the Källen-Lehmann representation of the propagator leads to the interpretation that the propagation of free-fields in a quantized spacetime is analogous to propagation of interacting fields in a continuous spacetime.

NON SCALE-INVARIANT ISOCURVATURE PERTURBATIONS PRODUCED IN THE POWER-LAW INFLATION

1991 ◽  
Vol 06 (32) ◽  
pp. 2935-2945 ◽  
Author(s):  
MISAO SASAKI ◽  
BORIS L. SPOKOINY
Keyword(s):  
Scalar Field ◽  
Power Law ◽  
Quantum Fluctuation ◽  
Careful Analysis ◽  
Baryon Density ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Exact Form ◽  
Point Function ◽  
Scale Invariant

We present a careful analysis of non scale-invariant insocurvature perturbation produced in the power-law inflation. We first derive the exact form of the two-point function for a massless scalar field φ in a power-law background, a (t) ∝ t1+n (n≫1). For generality, we allow the scalar field to have a small nonminimal coupling to gravity, ~ ξφ2 R (|ξ|≪1). We then regard φ as an axion-like field whose quantum fluctuation gives rise to an isocurvature perturbation with its amplitude proportional to a trigonometric function of φ. As a concrete example, we consider the case when φ is a Majoron and the baryon density fluctuation is produced in proportion to sin (φ/f) where f is the symmetry breaking scale. We find the resulting spectrum of the (baryon) isocurvature perturbation depends very much on the sign of n*, where 1/n* = 1/n + ξ. For n* > 0, corresponding to an infrared unstable scalar field, the spectrum is white noise on large scales and almost scale-invariant on small scales. On the other hand, for n* < 0, corresponding to an infrared stable scalar field, the spectrum is almost scale-invariant on all the scales.

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