Why no EOM in QFT with higher than second order derivatives in time and space?

In summary, it is important for a Lagrangian in quantum field theory to not depend on higher order time and space derivatives of \phi because it ensures that the equation of motion is at most second order. Higher order EOMs can cause non-locality, which can be seen in the generation of a Taylor series expansion. Negative powers of momentum in the Lagrangian lead to non-locality, and even though a finite series of positive powers is still local, there are other physical reasons to avoid higher powers of momentum. This is why effective field theories may include higher derivative powers, but a field theory with any higher derivative terms is not renormalizable and therefore not a fundamental theory.
  • #1
Phiphy
16
1
When we write down a Lagragian for a quantum field theory, it is said that it should not depend on the second and higher order time and space derivatives of [tex]\phi[/tex], because we want the equation of motion(EOM) to be at most second order. Why is it so important. What trouble will a higher order EOM cuase in physics? Could anyone give me some examples? Thanks.
 
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  • #2
The tale says that the problem is non-locality, that is you'd generate a Taylor series expansion.
 
  • #3
Non-locality happens when there are negative powers of momentum in the lagrangian. A finite series of positive powers of momentum is still local. There must be some other physical reasons to rule out higher powers of momentum.
 
  • #4
why do negative powers of momentum yield non-locality and in what sense?
 
  • #5
Effective field theories sometimes have higher derivative powers. But, a field theory with any higher derivative terms will not be renormalizable and, so, would be expected not to be a fundamental theory.
 

1. Why do higher order derivatives in time and space lead to problems in QFT?

Higher order derivatives in time and space lead to problems in QFT because they introduce non-locality in the theory. This means that the equations of motion at a particular point in space and time are affected by the values of the fields at distant points and times, making it difficult to calculate and predict the behavior of the system.

2. Can higher order derivatives be included in QFT at all?

Yes, higher order derivatives can be included in QFT, but they can only be included in a very limited way. This is because the equations of motion in QFT are required to be local, meaning that the fields at a particular point in space and time only depend on the values of the fields at that same point in space and time.

3. What are some specific issues that arise when trying to incorporate higher order derivatives in QFT?

Some specific issues that arise when trying to incorporate higher order derivatives in QFT include non-locality, causality violations, and unitarity violations. These issues can lead to inconsistencies and inconsistencies with experimental data, making it difficult to use higher order derivatives in QFT.

4. Are there any alternative theories that allow for higher order derivatives in QFT?

Yes, there are alternative theories such as non-local QFT and string theory that allow for higher order derivatives. However, these theories come with their own set of challenges and have not yet been fully developed or tested.

5. What are some potential implications of not being able to include higher order derivatives in QFT?

The inability to include higher order derivatives in QFT could limit our understanding and ability to accurately describe certain physical phenomena. It could also affect the accuracy and predictive power of the theory, making it more difficult to make precise calculations and predictions. Additionally, it could hinder our ability to unify different areas of physics, such as gravity and quantum mechanics.

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