# Give mass to a massless scalar field in 1+1, Higgs like?

1. May 24, 2015

### Spinnor

Is it possible to have a free massless scalar field in 1+1 spacetime and then add another field of the right type which interacts with some adjustable strength with the massless field to give mass to the massless field? Is there a Higgs-like mechanism in 1+1 spacetime?

Thanks for any help!

2. May 24, 2015

### thierrykauf

Have you looked at the Mermin Wagner theorem?

3. May 25, 2015

### Spinnor

Thank you! From the wiki article,

"
In quantum field theory and statistical mechanics, the Mermin–Wagner theorem (also known as Mermin–Wagner–Hohenberg theorem or Coleman theorem) states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions d ≤ 2. Intuitively, this means that long-range fluctuations can be created with little energy cost and since they increase the entropy they are favored.

This is because if such a spontaneous symmetry breaking occurred, then the corresponding Goldstone bosons, being massless, would have an infrared divergent correlation function.

The absence of spontaneous symmetry breaking in d ≤ 2 dimensional systems was rigorously proved by Sidney Coleman (1973) in quantum field theory and by David Mermin, Herbert Wagner and Pierre Hohenberg in statistical physics. That the theorem does not apply to discrete symmetries can be seen in the two-dimensional Ising model. "