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

• Spinnor
In summary, the Mermin-Wagner theorem states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with short-range interactions in dimensions d ≤ 2. This is due to the fact that long-range fluctuations are favored and would lead to an infrared divergent correlation function. However, this theorem does not apply to discrete symmetries, as seen in the two-dimensional Ising model.
Spinnor
Gold Member
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!

Have you looked at the Mermin Wagner theorem?

Spinnor
thierrykauf said:
Have you looked at the Mermin Wagner theorem?

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. "

thierrykauf

## 1. How can a massless scalar field be given mass in 1+1 dimensions?

In 1+1 dimensions, a massless scalar field can be given mass through a process known as spontaneous symmetry breaking. This involves the introduction of a new scalar field, called the Higgs field, which interacts with the massless scalar field and gives it mass through the Higgs mechanism.

## 2. What is the significance of giving mass to a massless scalar field in 1+1 dimensions?

Giving mass to a massless scalar field in 1+1 dimensions has significant implications for theories in theoretical physics, particularly in the field of particle physics. It allows for the explanation of how particles acquire mass and plays a crucial role in the Standard Model of particle physics.

## 3. Is the process of giving mass to a massless scalar field in 1+1 dimensions similar to the Higgs mechanism in 3+1 dimensions?

Yes, the process of giving mass to a massless scalar field in 1+1 dimensions is similar to the Higgs mechanism in 3+1 dimensions. Both involve the introduction of a new scalar field that interacts with the massless scalar field to give it mass.

## 4. Can the Higgs-like mechanism be applied to other fields besides the massless scalar field in 1+1 dimensions?

Yes, the Higgs-like mechanism can be applied to other fields besides the massless scalar field in 1+1 dimensions. In fact, it has been successfully applied to other fields in various theories, such as the electroweak theory and supersymmetry theories.

## 5. How does the Higgs-like mechanism affect the behavior of the massless scalar field in 1+1 dimensions?

The Higgs-like mechanism changes the behavior of the massless scalar field in 1+1 dimensions by introducing a new interaction that gives the field mass and breaks its initial symmetry. This results in the formation of a non-zero vacuum expectation value and the appearance of a massive scalar particle, known as the Higgs boson.

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