# What is the definition of the transverse mode in QFT?

• I
• Silviu
In summary, the conversation discussed the definition of the transverse component of a vector field in QFT. It was stated that any vector field can be written as the sum of its transverse component and a derivative term, and that the condition for the transverse component is that its derivative is equal to zero. The conversation also mentioned the notation used in the context and clarified a possible error in the notation.
Silviu
Hello! I am reading some QFT and at a point I read that any vector field (here we are working with massive spin 1 particles) can be written as: $$A_\mu(x)=A^T_\mu(x)+\partial_\mu\pi(x)$$ with $$\partial_\mu A^T_\mu(x)=0$$ They don't talk about notation, but from the context I understand that ##A^T_\mu(x)## is the transverse component of ##A_\mu(x)##. Is ##\partial_\mu A^T_\mu(x)=0## the definition of the transverse component? And if so, why? Thank you!

Silviu said:
Hello! I am reading some QFT and at a point I read that any vector field (here we are working with massive spin 1 particles) can be written as: $$A_\mu(x)=A^T_\mu(x)+\partial_\mu\pi(x)$$ with $$\partial_\mu A^T_\mu(x)=0$$ They don't talk about notation, but from the context I understand that ##A^T_\mu(x)## is the transverse component of ##A_\mu(x)##. Is ##\partial_\mu A^T_\mu(x)=0## the definition of the transverse component? And if so, why? Thank you!
Yes, that's the definition. Note that you meant ##\partial^\mu A^T_\mu(x)=0##.

nrqed said:
Note that you meant ##\partial^\mu A^T_\mu(x)=0##.
He might not be if he is reading Schwartz's QFT book. If I remember correctly, Schwartz starts by saying that he takes it for granted that students know that when summation inices appear one is covariant and the other contravariant and therefore puts all indices as subindices.

nrqed

## 1. What is the transverse mode of a field?

The transverse mode of a field refers to the spatial distribution of the field's electric and magnetic components. In other words, it describes the pattern of oscillations of the field in a plane perpendicular to the direction of propagation.

## 2. How is the transverse mode of a field different from the longitudinal mode?

The transverse mode is characterized by oscillations perpendicular to the direction of propagation, while the longitudinal mode is characterized by oscillations parallel to the direction of propagation.

## 3. What are the different types of transverse modes?

There are two main types of transverse modes: TE (Transverse Electric) and TM (Transverse Magnetic) modes. TE modes have a zero electric field component along the direction of propagation, while TM modes have a zero magnetic field component along the direction of propagation.

## 4. How are transverse modes important in optics?

Transverse modes play a crucial role in the propagation of light in optical systems. The spatial distribution of the electric and magnetic components of the field determines the shape and size of the light beam, and can also affect its intensity and polarization.

## 5. How can the transverse mode of a field be controlled?

The transverse mode of a field can be controlled by changing the geometry and boundary conditions of the medium in which it is propagating. For example, in a laser cavity, the size and shape of the cavity can be adjusted to select a specific transverse mode. Similarly, using optical components such as lenses and mirrors, the transverse mode of a light beam can be manipulated.

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