Quantum triviality

^{[ original research? ]}

In physics, quantum triviality is the phenomenon in which a classical theory that describes interacting particles, when quantized, becomes a quantum field theory that describes noninteracting free particles. Charge screening can restrict the value of the observable charges that appear in the theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. ^{[1] [2]}

Triviality and the renormalization group

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 the work of Wilson. 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.^{[ citation needed ]}

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.

Historical background

The first evidence of possible triviality of quantum field theories was obtained in the context of quantum electrodynamics by Lev Landau, Alexei Abrikosov, and Isaak Khalatnikov ^{[5] [6] [7]} who found the following relation between the observable charge g_obs and 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 (by Murray Gell-Mann and Francis E. Low)

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

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 g(μ) relies on the appearance of the function β(g). According to the classification by Nikolay Bogolyubov and Dmitry Shirkov, ^[8] there are three qualitatively different situations:

1. if ${\displaystyle \beta (g)}$ has a zero at the finite value g^*, then growth of g is saturated, i.e. ${\displaystyle g(\mu )\to g^{*}}$ for ${\displaystyle \mu \to \infty }$;
2. if ${\displaystyle \beta (g)}$ is non-alternating and behaves as ${\displaystyle \beta (g)\propto g^{\alpha }}$ with ${\displaystyle \alpha \leq 1}$ for large ${\displaystyle g}$, then the growth of ${\displaystyle g(\mu )}$ continues to infinity;
3. if ${\displaystyle \beta (g)\propto g^{\alpha }}$ with ${\displaystyle \alpha >1}$ for large ${\displaystyle g}$, then ${\displaystyle g(\mu )}$ is divergent at finite value ${\displaystyle \mu _{0}}$ and the real Landau pole arises: the theory is internally inconsistent due to indeterminacy of ${\displaystyle g(\mu )}$ for ${\displaystyle \mu >\mu _{0}}$.

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.

Higgs triviality

Concerns regarding quantum triviality have been relevant to the physics of the Higgs boson. A comparatively simple toy model of the "Higgs sector" of the Standard Model is provided by φ⁴ theory, i.e., a scalar field theory whose Lagrangian includes an interaction term that is quartic in the field φ. It is conjectured, based on considerable evidence, that this theory is "trivial": if the regularization cutoff is taken to zero distance or infinite energy, the theory will describe only free particles. Using this as a model for the Higgs boson, the conjectured triviality implies that the cutoff cannot be removed. Because there is an upper bound on how high the cutoff energy can be, there is an upper bound on the interaction strength, which ultimately implies an upper bound on the mass of the Higgs boson. ^[9]

See also

• Gaussian fixed point
• Hierarchy problem

References

1. Zinn-Justin, Jean (2007). Phase Transitions and Renormalization Group. Oxford University Press. p. 323. ISBN   978-0-19-922719-8.
2. Huang, Kerson (2010) [1998]. Quantum Field Theory: From Operators to Path Integrals. Wiley. pp. 255–256. ISBN   978-3-527-40846-7.
3. Kadanoff, L. P. (1966). "Scaling laws for Ising models near ${\displaystyle T_{c}}$". Physics Physique Fizika . 2: 263. doi:10.1103/PhysicsPhysiqueFizika.2.263.
4. Wilson, K. G. (1975). "The renormalization group: critical phenomena and the Kondo problem". Rev. Mod. Phys. 47 (4): 773. doi:10.1103/RevModPhys.47.773.
5. L. D. Landau; A. A. Abrikosov; I. M. Khalatnikov (1954). "On the Elimination of Infinities in Quantum Electrodynamics". Doklady Akademii Nauk SSSR . 95: 497.
6. L. D. Landau; A. A. Abrikosov & I. M. Khalatnikov (1954). "Asymptotic Expression for the Green's Function of the Electron in Quantum Electrodynamics". Doklady Akademii Nauk SSSR . 95: 773.
7. L. D. Landau; A. A. Abrikosov & I. M. Khalatnikov (1954). "Asymptotic Expression for the Green's Function of the Photon in Quantum Electrodynamics". Doklady Akademii Nauk SSSR . 95: 1177.
8. N. N. Bogoliubov; D. V. Shirkov (1980). Introduction to the Theory of Quantized Fields (3rd ed.). John Wiley & Sons. ISBN   978-0-471-04223-5.
9. Siefert, Johannes; Wolff, Ulli (2014). "Triviality of φ⁴₄ theory in a finite volume scheme adapted to the broken phase" (PDF). Physics Letters B. 733: 11–14. doi:10.1016/j.physletb.2014.04.013.