QFT : Why do tensors in lagrangian densities contract?

In summary, the general rule behind why any given Lagrangian in QED/QCD shows that vectors and tensors contract indices is because the action must be a scalar quantity. This applies to examples such as F_{\mu\nu}F^{\mu\nu} in QED, A_\mu A^\mu in Proca, and G^{\alpha}_{\mu\nu} G^{\mu\nu}_{\alpha} in QCD. While a non-real Lagrangian like B^{\mu\nu}B^{\mu}_{\nu} may not necessarily be gauge invariant, it still follows this rule. Additionally, B^{\mu}B_{\nu} is not a 16-term scalar
  • #1
Hepth
Gold Member
464
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What is the general rule behind why for any given lagrangian (QED/QCD show this) that any vectors or tensors contract indices? I know it must be something simple, but I just can't think of it offhand.
QED :
[tex]
F_{\mu\nu}F^{\mu\nu}
[/tex]
Proca (massive vector):
[tex]
A_\mu A^\mu
[/tex]
QCD :
[tex]
G^{\alpha}_{\mu\nu} G^{\mu\nu}_{\alpha}
[/tex]

Like could I imagine some non-real lagrangian that is [tex]B^{\mu\nu}B^{\mu}_{\nu}[/tex]
without worrying about gauge invariance?EDIT: its that the action has to be a scalar quantity, isn't it?
REEDIT: Ah its still a scalar though, just not NECESSARILY invariant.

Well then what about
[tex]
B^{\mu}B_{\nu}
[/tex]
so that you still get some 16 term scalar, but its not a similar-indice contraction.
 
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  • #2
That's not a "16-term scalar" (which I don't think even makes sense). That's a tensor.

You answered your own question with "the action has to be a scalar".
 
  • #3
Ok, just making sure.
 

Related to QFT : Why do tensors in lagrangian densities contract?

1. Why are tensors used in lagrangian densities?

Tensors are used in lagrangian densities because they allow us to mathematically describe the dynamics of a system in terms of its fundamental quantities, such as energy and momentum. In quantum field theory, these quantities are represented by tensors, which are mathematical objects that have the ability to change under certain operations. By using tensors in lagrangian densities, we can accurately model and predict the behavior of a physical system.

2. What is the purpose of contracting tensors in lagrangian densities?

The purpose of contracting tensors in lagrangian densities is to mathematically combine multiple tensors in a specific way to represent a physical phenomenon. This contraction is necessary in order to accurately describe the interactions between particles in a quantum field theory. By contracting tensors, we can simplify the mathematical representation of a system and make calculations more manageable.

3. How do tensors in lagrangian densities relate to symmetries?

Tensors in lagrangian densities play a crucial role in understanding symmetries in quantum field theory. Symmetries are mathematical operations that leave a physical system unchanged, and they are represented by tensors in lagrangian densities. By studying how tensors transform under these operations, we can gain a deeper understanding of the symmetries present in a system and use them to make predictions about its behavior.

4. Can tensors in lagrangian densities be manipulated?

Yes, tensors in lagrangian densities can be manipulated using mathematical operations such as contraction and tensor products. These manipulations are necessary in order to accurately describe the dynamics of a system and make predictions about its behavior. However, it is important to note that these manipulations must adhere to certain rules and principles in order to maintain the physical meaning of the tensors.

5. How are tensors in lagrangian densities related to the fundamental forces?

Tensors in lagrangian densities are closely related to the fundamental forces in nature. In quantum field theory, the interactions between particles are described by the exchange of force-carrying particles. These interactions are mathematically represented by tensors in lagrangian densities. By studying how these tensors transform under certain operations, we can gain insights into the fundamental forces and their role in shaping the behavior of physical systems.

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