A renormalizable QFT cannot have higher derivatives. Effective (nonrenormalizable) theories can.ndung200790 said:Why the Lagrangian in QFT does not include high order derivative of field?
Ben Niehoff said:Also, classically speaking, higher derivatives in the Lagrangian will result in higher derivatives in the equations of motion. In particular, if an equation of motion has more than two time derivatives, it becomes difficult to make physical sense out of it (because it seems to require too much initial data to define a unique solution).
A. Neumaier said:Not really. A k-th order differential equation requires as initial conditions the field and the first k-1 derivatives
at the initial time. The case k=2 is nothing special here.
Ben Niehoff said:Yes, I'm aware of that. However, our vast experience with classical mechanics indicates that k=2 is indeed special. For example, for a classical system to be a Hamiltonian flow on phase space requires k=2.
Ben Niehoff said:Also, classically speaking, higher derivatives in the Lagrangian will result in higher derivatives in the equations of motion. In particular, if an equation of motion has more than two time derivatives, it becomes difficult to make physical sense out of it (because it seems to require too much initial data to define a unique solution).
A. Neumaier said:Not really. A k-th order differential equation requires as initial conditions the field and the first k-1 derivatives
at the initial time. The case k=2 is nothing special here.
Ben Niehoff said:Yes, I'm aware of that. However, our vast experience with classical mechanics indicates that k=2 is indeed special. For example, for a classical system to be a Hamiltonian flow on phase space requires k=2.
The one case I know of in which k>2, the radiation reaction on a moving, charged source, has issues with causality and runaway solutions, precisely because of the third time derivative that appears in the equation of motion.
These are classical situations, but I think deep down they have some relationship to the nonrenormalizability of higher-derivative actions. There is something fundamentally quirky about a physical system that requires information about higher time derivatives.
One reason for this is that the Lagrangian, which is a function used to describe the dynamics of a system, is derived from the principle of least action. This principle states that a system will take the path of least action, which is the path that minimizes the difference between the initial and final states. In QFT, this means that the Lagrangian must only depend on the first order derivatives of the field, as this is the minimum necessary to describe the dynamics of the system.
Higher order derivatives of the field can be important in certain situations, but they are not included in the Lagrangian for a few reasons. Firstly, including higher order derivatives would result in equations of motion that are higher order, making them more difficult to solve. Additionally, many physical systems can be accurately described using only the first order derivatives. Finally, there is a principle in physics known as Occam's razor, which states that the simplest explanation is often the best. In this case, including only the first order derivatives of the field is the simplest and most elegant way to describe the system.
Yes, the Lagrangian in QFT can include higher order derivatives if needed. However, this is typically only done in situations where it is necessary, such as in certain quantum field theories or when studying systems with higher order dynamics. In most cases, the first order derivative is sufficient to accurately describe the dynamics of the system.
The use of the Lagrangian in QFT has several advantages. Firstly, it allows for a more elegant and concise description of the system's dynamics compared to other methods. Additionally, the Lagrangian formalism is closely related to the Hamiltonian formalism, which is useful for studying the energy and momentum of a system. Finally, the use of the Lagrangian allows for the application of the principle of least action, which is a powerful tool for understanding the behavior of physical systems.
While the Lagrangian is a powerful tool in QFT, it does have some limitations. One limitation is that it may not be applicable to all systems, particularly those with highly nonlinear dynamics. Additionally, the Lagrangian formalism may not be the most efficient or convenient method for solving certain problems. In these cases, other methods, such as the Hamiltonian formalism, may be more useful. However, overall, the Lagrangian is a valuable and widely used tool in QFT that has greatly contributed to our understanding of the fundamental laws of nature.