Classical equivalent of scalar free field in QFT

[eoghan]

TL;DR

Which classical system has the same lagrangian of a free scalar field in QFT?

Hi there,

In QFT, a free scalar field can be represented by the lagrangian density
$$\mathcal{L} = \frac{1}{2}\left(\partial\phi\right)^2 - \frac{1}{2}m^2\phi^2$$

I would like to find a classical system that has the same lagrangian. If we consider the transversal motion of an elastic string, the lagrangian would be
$$\mathcal{L} = \frac{1}{2}\left[\left(\partial_t\phi\right)^2 - \left(\partial_x\phi\right)^2\right]$$
This is very similar to the lagrangian of the free field, except that there is a missing term $\frac{1}{2}m^2\phi^2$. Which classical force can give rise to this term? I was thinking of a local elastic force, that pulls each element of the string downwards. So basically it is as if the string was standing horizontal and each of its points are connected to the ground by vertical springs.

Is this correct?

 

[Demystifier]

A mass term appears in certain acoustic wave equations. See Eqs. (37)-(41) in https://arxiv.org/abs/1802.03267

 

[hilbert2]

Here's another one:

http://www1.unipa.it/girep2014/accepted-papers-proceedings/212_Faletic.pdf

 

[dextercioby]

Summary:: Which classical system has the same lagrangian of a free scalar field in QFT?

Hi there,

In QFT, a free scalar field can be represented by the lagrangian density
$$\mathcal{L} = \frac{1}{2}\left(\partial\phi\right)^2 - \frac{1}{2}m^2\phi^2$$

I would like to find a classical system that has the same lagrangian. If we consider the transversal motion of an elastic string, the lagrangian would be
$$\mathcal{L} = \frac{1}{2}\left[\left(\partial_t\phi\right)^2 - \left(\partial_x\phi\right)^2\right]$$
This is very similar to the lagrangian of the free field, except that there is a missing term $\frac{1}{2}m^2\phi^2$. Which classical force can give rise to this term? I was thinking of a local elastic force, that pulls each element of the string downwards. So basically it is as if the string was standing horizontal and each of its points are connected to the ground by vertical springs.

Is this correct?

Just to make things rigorous: the quoted Lagrangian density you gave is from classical field theory. QFT uses the so-called "quantum fields", whose multiplication (and multiplication of space-time derivatives) is normally ill-defined.

So the classical Lagrangian is useful in QFT in two ways.
1. For the Feynman path integral formalism.
2. Its classical solutions of the EOM lead to quantized free fields. Putting classical Poisson brackets to quantum brackets (commutators of operators in a Fock space) is the key ingredient of quantization.

 

[vanhees71]

3. To figure out the canonical field momenta for "canonical quantization", leading to the operator formalism.

 

[eoghan]

Thank you very much for all your answers :)