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We construct a quantum theory of free scalar fields in (1+1)-dimensions based on the deformed Heisenberg algebra

Physics in extremely high energy scales is particularly of interest to particle physicists. It is now a well-known issue that gravity induces uncertainty in measurement of physical quantities. By now, string theory is one of the most successful theoretical frameworks which overcomes the difficulty of ultraviolet divergences in quantum theory of gravity. Incorporation of gravity in quantum field theory leads naturally to an effective cutoff (a minimal measurable length) in the ultraviolet regime. Therefore, if we construct a field theory which captures some stringy nature and/or includes stringy corrections, then it would play a crucial role in investigation of physics at high energy scales towards the Planck scale. Some of the stringy corrections appear as higher order polynomials of momentum leading to modified dispersion relations (see e.g., [

We investigate the quantization of free scalar field based on the deformed algebra (

In ordinary quantum mechanics, the standard Heisenberg uncertainty principle (HUP) is given by

At energies much below the Planck energy, the extra term in the right hand side of (

Now it is easy to show that

Magueijo and Smolin have shown that in the context of the Doubly Special Relativity a test particle’s momentum cannot be arbitrarily imprecise and therefore there is an upper bound for momentum fluctuation [

We start with the following deformed Heisenberg algebra:

This is not equal to the original function for arbitrary function

Now for these functions the transformation (

This restriction is nothing but the condition that the state

In this section we construct a quantum field theory with a GUP (that admits both a minimal length and a maximal momentum) in

The generalization of the Heisenberg algebra to higher dimensions where rotation symmetry is preserved and there are both a minimal length and a maximal momentum is (see [

Incorporation of quantum gravity effects in quantum mechanics and also quantum field theory leads to the existence of natural cutoffs on position and momentum measurements of test particles. These natural cutoffs regularize the underlying field theory in a phenomenologically viable manner. Quantum field theory of scalar fields in the presence of minimal length has been studied in literature (see [

K. Nozari work is financially supported by the Research Deputy of the Islamic Azad University, Sari Branch, Sari, Iran.