# Classical equivalent of scalar free field in QFT

• I
• eoghan
In summary, the conversation discusses the use of a classical system with the same lagrangian as a free scalar field in QFT. It is suggested that the transversal motion of an elastic string has a similar lagrangian, but with a missing term that could be accounted for by a local elastic force. However, it is noted that this lagrangian is from classical field theory and may not be directly applicable to QFT.
eoghan
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?

Delta2
eoghan said:
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

## 1. What is the classical equivalent of a scalar free field in quantum field theory?

The classical equivalent of a scalar free field in quantum field theory is a classical field theory that describes the same physical system as the quantum field theory. It is often used as a starting point for understanding the behavior of quantum fields.

## 2. How does the classical equivalent of a scalar free field differ from the quantum field theory?

The main difference between the classical equivalent and the quantum field theory is that the classical theory treats the field as a continuous function of space and time, while the quantum theory describes the field as a collection of discrete particles.

## 3. What is the role of the classical equivalent in quantum field theory?

The classical equivalent serves as a useful tool for understanding the behavior of quantum fields and for making predictions about their interactions. It also provides a way to connect the concepts of classical and quantum mechanics.

## 4. How is the classical equivalent of a scalar free field calculated?

The classical equivalent of a scalar free field is typically calculated using the classical Hamiltonian formalism, which involves finding the Lagrangian of the system and then using the Euler-Lagrange equations to determine the equations of motion for the field.

## 5. What are the applications of the classical equivalent of a scalar free field in QFT?

The classical equivalent of a scalar free field has various applications in quantum field theory, including in the study of phase transitions, the calculation of scattering amplitudes, and the understanding of quantum field dynamics in curved spacetime.

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