Abstract
The relation between the subtracted Green’s functions with different choices of subtraction point in the \(\phi ^4_4\) model. The running coupling constant. Functional equations of the renormalization group. Differential renormalization group equations of the Gell-Mann–Low and the Callan–Symanzik type. The \(\beta \) function. Reliability of the perturbative approximations. The phenomenon of dimensional transmutation in renormalized quantum field theory.
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Notes
- 1.
The proof can be found in, e.g., [8].
- 2.
In the presented approach to the renormalization group they are just identities which follow from the definitions of \(\underline{\lambda }, \, \underline{m}\) and \(z_3\). Nevertheless, we shall call them equations as in most textbooks.
- 3.
In the natural units (\(\hbar =1, \, c=1\)) the field \(\phi (x)\) has the dimension \(\text {cm}^{-1}\), and the vacuum state vector \(| 0 \rangle \) is dimensionless, hence \([G^{(n)}] = \text {cm}^{-n}\). The Fourier transform changes the dimension by \(+4n\). Therefore, \([\tilde{G}^{(n)}] = \text {cm}^{+3n} = [m_0]^{-3n}\).
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Arodź, H., Hadasz, L. (2017). The Renormalization Group. In: Lectures on Classical and Quantum Theory of Fields. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-55619-2_9
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DOI: https://doi.org/10.1007/978-3-319-55619-2_9
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