Abstract
In this final chapter, we continue investigating the theory of deformed scalar field uncovering its simplest quantum aspects.
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Notes
- 1.
The non-commutative delta function \(\hat{\delta }(\hat{x},\hat{y})\) also satisfies:
$$\begin{aligned} \widehat{\int _{\hat{x},\hat{y}}}\hat{\delta }(\hat{x},\hat{y})=1 \ . \end{aligned}$$ - 2.
Given the propagation law \(\phi (x)=-\int d^{4}y \ \varDelta _{F}^{\kappa }(x-y) \ \sqrt{1+\square /\kappa ^{2}} \ J(y)\) we can take the formal series expansion in powers of the d’Alembertian for the star product term \(\phi (x)=-\int d^{4}y \ \sum _{n=0}^{\infty }a_{n}\varDelta _{F}^{\kappa }(x-y) \ \square ^{n} J(y)\) which, after integrating by parts becomes \(\phi (x)=-\int d^{4}y \ \sum _{n=0}^{\infty } a_{n} \square ^{n}\varDelta _{F}^{\kappa }(x-y) \ J(y)\).
- 3.
Here and below, we ignore a possible phase factor that may be present in the definition.
- 4.
It is, of course, a matter of pure convention, which is particle and which is antiparticle.
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Arzano, M., Kowalski-Glikman, J. (2021). Free Quantum Fields and Discrete Symmetries. In: Deformations of Spacetime Symmetries. Lecture Notes in Physics, vol 986. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-63097-6_7
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