43.6 Interacting scalar field theories: some comments

43.6 Interacting scalar field theories: some comments

Our discussion so far has dealt purely with a theory of non-interacting quanta, so this theory is called a quantum theory of free fields. The field however can be used to introduce interactions between these quanta, interactions which are local in space. The simplest such theory is the one given by adding a quartic term to the Hamiltonian, taking

$$\hat{H}={\int }_{{\mathbf{R}}^3}:\frac{1}{2}(\hat{\Pi }(\mathbf{x}{)}^2+(\nabla \hat{\Phi }(\mathbf{x}){)}^2+m^2\hat{\Phi }(\mathbf{x}{)}^2)+\lambda \hat{\Phi }(\mathbf{x}{)}^4:d^3\mathbf{x}$$

This interacting theory is vastly more complicated and much harder to understand than the non-interacting theory. Among the dificult problems that arise are:

• How should one make sense of the expression

$$:\hat{\Phi }(\mathbf{x}{)}^4:$$

since it is a product not of operators but of operator-valued distributions?

• How can one construct an appropriate state space on which the interacting Hamiltonian operator will be well-defined, with a well-defined ground state?

Quantum field theory textbooks explain how to construct a series expansion in powers of $\lambda$ about the free field value $\lambda =0$ , by a calculation whose terms are labeled by Feynman diagrams. To get finite results, cutofs must first be introduced, and then some way found to get a sensible limit as the cutof is removed (this is the theory of “renormalization”). In this manner finite results can be found for the terms in the series expansion, but the expansion is not convergent, giving only an asymptotic series (for fixed $\lambda$, no matter how small, the series will diverge at high enough order).

For known calculational methods not based on the series expansion, again a cutof must be introduced, making the number of degrees of freedom finite.

For a fixed ultraviolet cutof, corresponding physically to only allowing fields with momentum components smaller than a given value, one can construct a sensible theory with non-trivial interactions (e.g., scattering of one particle by another, which does not happen in the free theory). This gives a sensible theory, for momenta far below the cutof. However, it appears that, for three or more spatial dimensions, removal of the cutof will always in the limit give back the non-interacting free field theory. There is thus no known continuum relativistic quantum field theory of scalar fields, other than free field theory, that can be constructed in this way.

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