One of the simplest QFT model is $\lambda \phi^4$ theory.

$$ S = \int dt d^dx\; \left(\frac{1}{2}(\partial_\mu \phi)^2 - \frac{1}{4!}\lambda \phi^4 \right) $$

Equation of motion:

$$ \Box \phi = -\frac{1}{3!}\phi^3 $$

The equation is non-linear.

What is known about solutions of such e.o.m.?

What is physical interpretation of such solutions? How to match these solutions to spectrum of theory? Is it possible to relate some solutions to particles?

Note that my question about any dimension of space time. I think, that there are a lot about $d+1=2$ and $d+1 =4$, but unfortunately I never face with discussion of such questions.

  • $\begingroup$ To clarify: Are you asking about solution of the classical equation of motion, disregarding quantum field theory's operator-product issues? And are you asking about the case of Lorentzian signature or Euclidean signature? $\endgroup$ Oct 10 '20 at 23:54
  • $\begingroup$ @ChiralAnomaly About classical equations. In Lorentzian signature. (If you know about Euclidean, please also give me information) $\endgroup$
    – Nikita
    Oct 11 '20 at 0:34

There is one rather famous solution, called Fubini instanton by the name of it founder, discovered in 1976. https://link.springer.com/article/10.1007%2FBF02785664

The abstract claims:

A new approach to conformal invariant field theories is presented. The physical idea is to introduce a fundamental scale of hadron phenomena by means of a dilatation noninvariant vacuum state in the framework of a scale-invariant Lagrangian field theory. A new unconventional feature is that this programme can only be carried out if the «vacuum» state is not translation invariant. The «vacuum» is still invariant under a 10-parameter subgroup of the full conformal group but this subgroup does not coincide with the Poincaré group. A physical interpretation based on an appropriate averaging on an infinite ensemble of equivalent «vacuum» states allows to preserve energy momentum conservation, which is defined within a thermodynamical framework.

This solution describes the bounce solution, representing decay of the vacuum state. There are generalisations to curved space - https://arxiv.org/abs/1204.1521. However, these solutions were derived for $3+1$ case, and if you are asking about arbitrary dimension, I am unaware of any results there.

Moreover, there is a also a way, to obtain a solution of this equation in terms of formal series, which was invented by Rosly and Selivanov in 1996. Represent a solution of equation of motion in terms of formal series: $$ \phi(x) = \sum_i \eta_i \phi_i e^{i k_i x} + \sum_{i, j} \eta_i \eta_j \phi_{ij} e^{i (k_i + k_j) x} + \ldots $$ Where $\eta_i$ are nilpotent non-commuting variables, such that $\eta_i^2 = 0$. The substitution in the equation of motion $\Box \phi = -\frac{1}{3!} \phi^3$ leads to the following recursion relationships: \begin{align} \phi_{i j} &= 0 \\ (k_i + k_j + k_k)^2 \phi_{i j k} &= - \frac{1}{3!} \phi_i \phi_j \phi_k \end{align} Where we fix initially $\phi_i$. And so on. Each coefficient for order say $n$ in these series would be a sum of all amplitudes with $n$-external particles, and the off-shell leg.


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