Recent theorems

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Some recent ideas are generalized from four dimensions to the general dimension $n$. In quantum field theory, two terms of the trace anomaly in external gravity, the Euler density $G_n$ and $\Box^{n/2-1}R$, are relevant to the problem of quantum irreversibility. By adding the divergence of a gauge-invariant current, $G_n$ can be extended to a new notion of Euler density, linear in the conformal factor. We call it pondered Euler density. This notion relates the trace-anomaly coefficients $a$ and $a’$ of $G_n$ and $\Box^{n/2-1}R$ in a universal way ($a=a’$) and gives a formula expressing the total RG flow of a as the invariant area of the graph of the beta function between the fixed points. I illustrate these facts in detail for $n=6$ and check the prediction to the fourth-loop order in the $\phi^3$-theory. The formula of quantum irreversibility for general n even can be extended to $n$ odd by dimensional continuation. Although the trace anomaly in external gravity is zero in odd dimensions, I show that the odd-dimensional formula has a predictive content.


Nucl.Phys. B567 (2000) 331-359 | DOI: 10.1016/S0550-3213(99)00479-4


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14B1 D. Anselmi

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Last update: May 9th 2015, 230 pages

Contents: Preface | 1. Functional integral | 2. Renormalization | 3. Renormalization group | 4. Gauge symmetry | 5. Canonical formalism | 6. Quantum electrodynamics | 7. Non-Abelian gauge field theories | Notation and useful formulas | References

Course on renormalization, taught in Pisa in 2015. (More chapters will be added later.)