M.E. Irizarry-Gelpí

Physics impostor. Mathematics interloper. Husband. Father.

Non-Polynomial Scalar Interactions


A common interaction in quantum field theory is a non-linear term in the lagrangian of the form

$$ g_{n} \vert \phi \vert^{2} A^{n}, $$

with \(n\) being an integer greater or equal to one. According to dimensional analysis, in \(D\) spacetime dimensions the coupling \(g_{n}\) has a mass dimension given by

$$ \Delta = \frac{4 +2n - nD}{2}. $$

The cases when \(\Delta = 0\) are very important:

$$ (D = 3, \, n = 4), \qquad (D = 4, \, n = 2), \qquad (D = 6, \, n = 1). $$

For many important reasons, any serious quantum field theory should live in a spacetime with integer number of dimensions, and only have positive integer powers of fields in the lagrangian. However, we can relax a bit and ask two different questions:

  1. What values of \(n\) give \(\Delta = 0\) for given value of \(D\)?
  2. What values of \(D\) give \(\Delta = 0\) for given value of \(n\)?

If \(\Delta = 0\), then you have

$$ n = \frac{4}{D - 2}. $$

Obviously, \(D = 2\) is problematic. The first non-trivial case is \(D = 5\), giving \(n = 4/3\). Beyond six spacetime dimensions, you get:

$$ (D, n) = (7, 4/5), \quad (8, 2/3), \quad (9, 4/7), \quad (10, 1/2), \quad (11, 4/9), \quad ... $$

On the other hand, if \(\Delta = 0\), then you also have

$$ D = 2 + \frac{4}{n}. $$

Again \(D = 2\) is problematic. The first non-trivial case is \(n = 3\), giving \(D = 10/3\). Note that

$$ 3 < \frac{10}{3} < 4. $$

There are spaces with fractional number of dimensions: fractals. So I am curious if one could define a field theory living in a space with a rational number of dimensions.

Beyond the sixth-order interaction, you get:

$$ (D, n) = (14/5, 5), \quad (8/3, 6), \quad (18/7, 7), \quad (5/2, 8), \quad (22/9, 9), \quad ...$$

A coupling with vanishing mass dimension is a good candidate for a conformal field theory.