# QFT - rewriting a conserved quantity

• center o bass
In summary, the conversation is about the energy momentum tensor and how it can be rewritten as (Jμ)ρv using a manipulation involving the antisymmetric tensor ωμν. The motivation behind this manipulation is to make it resemble a cross product.
center o bass
Hey! I'm trying to learn QFT now and I'm currently reading David Tong's online lectures;

http://www.damtp.cam.ac.uk/user/tong/qft/one.pdf.

At page 17 finds the conserved current

$$j^\mu = - \omega^\rho_{\ \nu} T^{\mu}_{\ \rho} x^\nu$$

where i have understood T to be the energy momentum tensor. He further states that it can be rewritten as

$$(J^\mu)^{\sigma \rho} = x^\rho T^{\mu \sigma} - x^\sigma T^{\mu \rho}.$$

I am not that good manipulating tensors yet and my question is how one goes about showing this, step by step.

Since ωμν is antisymmetric, ωμν = ½(ωμv - ω). We can therefore write (1.54) as
jμ = - ½(ωρv - ω)Tμρxν.
In the second term, switch the index labels ρ and v:
jμ = ½ωρv(Tμvxρ - Tμρxv)
He then defines (Jμ)ρv to be the quantity in parentheses.

Bill_K said:
Since ωμν is antisymmetric, ωμν = ½(ωμv - ω). We can therefore write (1.54) as
jμ = - ½(ωρv - ω)Tμρxν.
In the second term, switch the index labels ρ and v:
jμ = ½ωρv(Tμvxρ - Tμρxv)
He then defines (Jμ)ρv to be the quantity in parentheses.

Thanks! And there has also been one applied lowering operator and one raising operator?

What is the motivation behind doing this manipulation btw? To make it look kind of like a cross product?

## 1. What is QFT and how does it relate to rewriting a conserved quantity?

Quantum field theory (QFT) is a theoretical framework used to describe the behavior of subatomic particles and their interactions. In QFT, conserved quantities, such as energy and momentum, can be rewritten in terms of field operators, which are mathematical quantities that represent the fundamental particles. This allows for a better understanding and description of the dynamics and symmetries of particles at the quantum level.

## 2. Why is it important to rewrite conserved quantities in QFT?

Rewriting conserved quantities in QFT allows for a more accurate and comprehensive understanding of the behavior of subatomic particles. It also helps to reveal underlying symmetries and patterns, which can lead to new insights and predictions in the field of particle physics.

## 3. How is the conservation of energy and momentum expressed in QFT?

In QFT, the conservation of energy and momentum is expressed through the use of Noether's theorem, which states that for every continuous symmetry in a physical system, there exists a corresponding conserved quantity. This means that the conservation of energy and momentum can be attributed to the symmetries in the system.

## 4. Can conserved quantities be rewritten in terms of different field operators in QFT?

Yes, conserved quantities can be rewritten in terms of different field operators in QFT. This is because different field operators represent different particles with different properties, and therefore, different conserved quantities. This allows for a more versatile and adaptable approach to studying and understanding the behavior of particles in QFT.

## 5. How does the rewriting of conserved quantities in QFT impact our understanding of the universe?

The rewriting of conserved quantities in QFT has greatly impacted our understanding of the universe, particularly in the field of particle physics. It has allowed for the development of new theories and models, such as the Standard Model, which accurately describes the behavior of particles and their interactions. It has also led to the discovery of new particles and the prediction of their properties, expanding our knowledge of the fundamental building blocks of the universe.

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