Is V_{eff} truly the density of effective action in QFT?

In summary, the effective potential and the density of effective action are two distinct mathematical quantities in quantum field theory. The effective potential determines the energy of the system while the density of effective action contains information about the system's dynamics. They are related through a Legendre transformation but cannot always be used interchangeably. These tools are specific to certain types of quantum field theories and have applications in calculating physical quantities and studying phase transitions and the behavior of matter.
  • #1
ndung200790
519
0
Please teach me this:
Why do we not call V[itex]_{eff}[/itex]=[itex]\Gamma_{eff}[/itex]/VT (where [itex]\Gamma[/itex][itex]_{eff}[/itex] is the effective action, VT is spacetime volume) being density of effective action of field?
Thank you very much in advance.
 
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  • #2


I have just known(from Schoeder's QFT book) that V[itex]_{eff}[/itex]=v[itex]_{0}[/itex]([itex]\Phi[/itex][itex]_{cl}[/itex])[itex]\Phi[/itex][itex]^{4}_{cl}[/itex].Then this must be called effective potential,because it relates with interaction Phi-4(this case is an example) but must not be called density of action.Is that correct?
 

1. Why is the effective potential not called the density of effective action in quantum field theory?

The effective potential and the density of effective action are two different mathematical quantities that serve different purposes in quantum field theory. The effective potential is a function of the quantum field that determines the energy of the system, while the density of effective action is a functional of the quantum field that contains information about the dynamics of the system. Therefore, the two cannot be used interchangeably and are given different names to reflect their distinct roles in the theory.

2. What is the difference between the effective potential and the density of effective action?

The effective potential is a scalar function that describes the energy of a system as a function of the quantum field. It is related to the potential energy in classical mechanics and is used to calculate the ground state energy of a system. On the other hand, the density of effective action is a functional that contains information about the dynamics of the system, including the interactions between particles. It is used to calculate the scattering amplitudes and other observables in quantum field theory.

3. How is the effective potential related to the density of effective action?

The effective potential and the density of effective action are related through a Legendre transformation. This transformation allows us to switch between the two quantities and use them interchangeably in certain calculations. However, the two quantities have different physical interpretations and cannot always be used interchangeably.

4. Can the effective potential and the density of effective action be used to describe all quantum field theories?

No, the effective potential and the density of effective action are specific tools that are used in certain types of quantum field theories, such as scalar field theories. They cannot be applied to all quantum field theories, as different theories may require different mathematical tools to describe them.

5. What are some applications of the effective potential and the density of effective action in quantum field theory?

The effective potential and the density of effective action are important tools in studying the behavior of quantum fields. They are used to calculate various physical quantities, such as the ground state energy, scattering amplitudes, and particle interactions. They are also used in theoretical studies of phase transitions and the behavior of matter at high temperatures and densities.

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