Quantum triviality

Modern considerations of triviality are usually formulated in terms of the real-space renormalization group, largely developed by Kenneth Wilson and others. Investigations of triviality are usually performed in the context of lattice gauge theory. A deeper understanding of the physical meaning and generalization of the renormalization process, which goes beyond the dilatation group of conventional renormalizable theories, came from condensed matter physics. Leo P. Kadanoff's paper in 1966 proposed the "block-spin" renormalization group.[3] The blocking idea is a way to define the components of the theory at large distances as aggregates of components at shorter distances.

This approach covered the conceptual point and was given full computational substance[4] in Wilson's extensive important contributions. The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the Kondo problem, in 1974, as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and critical phenomena in 1971. He was awarded the Nobel prize for these decisive contributions in 1982.

In more technical terms, let us assume that we have a theory described by a certain function ${\displaystyle Z}$ of the state variables ${\displaystyle \{s_{i}\}}$ and a certain set of coupling constants ${\displaystyle \{J_{k}\}}$. This function may be a partition function, an action, a Hamiltonian, etc. It must contain the whole description of the physics of the system.

Now we consider a certain blocking transformation of the state variables ${\displaystyle \{s_{i}\}\to \{{\tilde {s}}_{i}\}}$, the number of ${\displaystyle {\tilde {s}}_{i}}$ must be lower than the number of ${\displaystyle s_{i}}$. Now let us try to rewrite the ${\displaystyle Z}$ function only in terms of the ${\displaystyle {\tilde {s}}_{i}}$. If this is achievable by a certain change in the parameters, ${\displaystyle \{J_{k}\}\to \{{\tilde {J}}_{k}\}}$, then the theory is said to be renormalizable. The most important information in the RG flow are its fixed points. The possible macroscopic states of the system, at a large scale, are given by this set of fixed points. If these fixed points correspond to a free field theory, the theory is said to be trivial. Numerous fixed points appear in the study of lattice Higgs theories, but the nature of the quantum field theories associated with these remains an open question.[2]

The first evidence of possible triviality of quantum field theories was obtained by Landau, Abrikosov, and Khalatnikov[5][6][7] by finding the following relation of the observable charge g_obs with the "bare" charge g₀,

${\displaystyle g_{\text{obs}}={\frac {g_{0}}{1+\beta _{2}g_{0}\ln \Lambda /m}}~,}$

(1)

where m is the mass of the particle, and Λ is the momentum cut-off. If g₀ is finite, then g_obs tends to zero in the limit of infinite cut-off Λ.

In fact, the proper interpretation of Eq.1 consists in its inversion, so that g₀ (related to the length scale 1/Λ) is chosen to give a correct value of g_obs,

${\displaystyle g_{0}={\frac {g_{\text{obs}}}{1-\beta _{2}g_{\text{obs}}\ln \Lambda /m}}~.}$

(2)

The growth of g₀ with Λ invalidates Eqs. (1) and (2) in the region g₀ ≈ 1 (since they were obtained for g₀ ≪ 1) and the existence of the "Landau pole" in Eq.2 has no physical meaning.

The actual behavior of the charge g(μ) as a function of the momentum scale μ is determined by the full Gell-Mann–Low equation

${\displaystyle {\frac {dg}{d\ln \mu }}=\beta (g)=\beta _{2}g^{2}+\beta _{3}g^{3}+\ldots ~,}$

(3)

which gives Eqs.(1),(2) if it is integrated under conditions g(μ) = g_obs for μ = m and g(μ) = g₀ for μ = Λ, when only the term with ${\displaystyle \beta _{2}}$ is retained in the right hand side.

The general behavior of ${\displaystyle g(\mu )}$ relies on the appearance of the function β(g). According to the classification by Bogoliubov and Shirkov,[8] there are three qualitatively different situations:

The latter case corresponds to the quantum triviality in the full theory (beyond its perturbation context), as can be seen by reductio ad absurdum. Indeed, if g_obs is finite, the theory is internally inconsistent. The only way to avoid it, is to tend ${\displaystyle \mu _{0}}$ to infinity, which is possible only for g_obs → 0.

As a result, the question of whether the Standard Model of particle physics is nontrivial remains a serious unresolved question. Theoretical proofs of triviality of the pure scalar field theory exist, but the situation for the full standard model is unknown. The implied constraints on the standard model have been discussed.[9][10][11] [12][13][14]

• Hierarchy problem