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eoghan

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- TL;DR 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?

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?