Journal ArticleDOI
Quantum field-theory models in less than 4 dimensions
Reads0
Chats0
TLDR
In this article, the authors studied the scalar Fermi interaction with anomalous dimensions in the zero-mass limit for space-time dimensions between 2 and 4, and showed that these dimensions are remarkably close to canonical except for the stress energy tensor.Abstract:
The scalar ${\ensuremath{\lambda}}_{0}{\ensuremath{\varphi}}^{4}$ interaction and the Fermi interaction ${G}_{0}{(\overline{\ensuremath{\psi}}\ensuremath{\psi})}^{2}$ are studied for space-time dimension $d$ between 2 and 4. An unconventional coupling-constant renormalization is used: ${\ensuremath{\lambda}}_{0}={u}_{0}{\ensuremath{\Lambda}}^{\ensuremath{\epsilon}}$ ($\ensuremath{\epsilon}=4\ensuremath{-}d$) and ${G}_{0}={g}_{0}{\ensuremath{\Lambda}}^{2\ensuremath{-}d}$, with ${u}_{0}$ and ${g}_{0}$ held fixed as the cutoff $\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$. The theories can be solved in two limits: (1) the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ where $\ensuremath{\varphi}$ and $\ensuremath{\psi}$ are fields with $N$ components, and (2) the limit of small $\ensuremath{\epsilon}$, as a power series in $\ensuremath{\epsilon}$. Both theories exhibit scale invariance with anomalous dimensions in the zero-mass limit. For small $\ensuremath{\epsilon}$, the fields $\ensuremath{\varphi}$, ${\ensuremath{\varphi}}^{2}$, and $\ensuremath{\varphi}{\ensuremath{\nabla}}_{{\ensuremath{\alpha}}_{1}}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\nabla}}_{{\ensuremath{\alpha}}_{n}}\ensuremath{\varphi}$ all have anomalous dimensions, except for the stress-energy tensor. These anomalous dimensions are calculated through order ${\ensuremath{\epsilon}}^{2}$; they are remarkably close to canonical except for ${\ensuremath{\varphi}}^{2}$. The ${(\overline{\ensuremath{\psi}}\ensuremath{\psi})}^{2}$ interaction is studied only for large $N$; for small $\ensuremath{\epsilon}$ it generates a weakly interacting composite boson. Both the ${\ensuremath{\varphi}}^{4}$ and ${(\overline{\ensuremath{\psi}}\ensuremath{\psi})}^{2}$ theories as solved here reduce to trivial free-field theories for $\ensuremath{\epsilon}\ensuremath{\rightarrow}0$. This paper is motivated by previous work in classical statistical mechanics by Stanley (the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit) and by Fisher and Wilson (the $\ensuremath{\epsilon}$ expansion).read more
Citations
More filters
Journal ArticleDOI
Bounding scalar operator dimensions in 4D CFT
TL;DR: In this article, a theory-independent inequality [phi(2)] 1 was derived for 4D conformal fixed points, where f(d) = 2 + O(root d - 1), which shows that the free theory limit is approached continuously.
Book
Foundations of Perturbative QCD
TL;DR: In this article, a systematic treatment of perturbative QCD is given, giving an accurate account of the concepts, theorems and their justification, giving strong motivations for the methods.
Journal ArticleDOI
The Asymptotic Safety Scenario in Quantum Gravity
Max Niedermaier,Martin Reuter +1 more
TL;DR: In this paper, a renormalizable quantum theory of the gravitational field is presented, which reconciles asymptotically safe couplings with unitarity, based on symmetry truncations and from truncated flow of the effective average action.
Journal ArticleDOI
Quantum field theory in the large N limit: a review
Moshe Moshe,Jean Zinn-Justin +1 more
TL;DR: In this article, a general scalar U ( φ 2 ) field theory for N large is presented, which is then applied to other issues such as tricritical behaviour and double scaling limit.
Book
Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory
TL;DR: In this article, the authors used random walk representations as a tool to derive correlation inequalities for critical-exponent theory and the consequences of these inequalities for the theory of critical phenomena and quantum field theory.