Lagrangian, scalar or pseudo-scalar?

In summary, the conversation discusses the requirement for a Lagrangian density to be a scalar under the spacetime symmetry group in quantum theory. The usual requirement is for the fields to transform irreducibly under the symmetry group, but in this case where the Lagrangian density is a pseudo-scalar, it would also be an eigenfunction of the parity operator. This topic is more relevant in quantum theory than classical physics, and the symmetry requirement for QFT in Minkowski spacetime is restricted to the Poincare group, CPT, and gauge invariance. Ultimately, the Lagrangian density should be a scalar with respect to all three combined or each taken separately.
  • #1
salparadise
23
0
Hi,

My question is. Can in principle, a Lagrangian density for some theory be a pseudo-scalar. Normally people say that the Lagrangian needs to be a scalar, but it case it is a pseudo-scalar it would also be a eigaen function of the parity operator.

This topic could well be on the classical physics section, but as this is more relevant in quantum theory I decided to place it here.

Thanks
 
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  • #2
The normal requirement is that the fields transform irreducibly under the spacetime symmetry group and that the lagrangian density is a scalar under the spacetime symmetry group. The symmetry requirement for QFT in Minkowski spacetime is restricted Poincare group plus CPT plus gauge invariance. The Lagrangian density should be a scalar wrt to the 3 altogether or each taken separately.
 

1. What is a Lagrangian?

A Lagrangian is a mathematical function used in physics to describe the dynamics of a system. It is based on the principle of least action, which states that the actual path of a system between two points is the one that minimizes the action, a quantity that combines the system's kinetic and potential energies.

2. What is the difference between a scalar and a pseudo-scalar?

A scalar is a physical quantity that only has magnitude, while a pseudo-scalar has both magnitude and direction. In other words, a scalar is a quantity that remains the same regardless of the coordinate system used, while a pseudo-scalar changes sign when the coordinate system is inverted.

3. How are Lagrangians used in physics?

Lagrangians are used to describe the motion of particles, fields, and other physical systems in classical mechanics, quantum mechanics, and general relativity. They provide a concise and elegant way to express the equations of motion for a system and allow for the use of powerful mathematical techniques to solve them.

4. Can Lagrangians be used in all physical systems?

Yes, Lagrangians can be used in all physical systems, as long as the system can be described by a set of coordinates and a potential energy function. However, they are most commonly used in systems with a finite number of degrees of freedom, such as particles or rigid bodies.

5. How do scalar and pseudo-scalar fields differ?

A scalar field is a physical quantity that has a single value at each point in space, while a pseudo-scalar field has a different value at each point depending on the orientation of the coordinate system. This means that a pseudo-scalar field can have different values at the same point in space, depending on the coordinate system used to measure it.

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