Spin of the sress-energy tensor in d=2

In summary, the spin of T^{zz} is 2, the spin of T^{\bar z \bar z} is -2, and the spin of $\Theta$ is 0.
  • #1
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If we suggest a generic quantum field theory and assume that the theory is Poincare-Invariant (i.e. the corresponding Ward-Identities are satisfied) than the stress energy tensor in two dimensions can be written in terms of complex coordinates [tex]z,\bar{z}[/tex]as
[tex]T^{zz}(z,\bar{z})=T^{00}-T^{11}-2iT^{10}[/tex]
[tex]T^{\bar{z}\bar{z}}(z,\bar{z})=T^{00}-T^{11}+2iT^{10}[/tex]
[tex]T^{z\bar{z}}(z,\bar{z})=T^{\bar{z}z}(z,\bar{z})=T^{00}+T^{11}\equiv -\Theta(z,\bar{z})[/tex]
My question is how to find the Spin of the components [tex]T^{zz},T^{\bar{z}\bar{z}},\Theta[/tex]. The authors of the paper I'm studying claim
[tex]ST^{zz}=2, ST^{\bar{z}\bar{z}}=-2 , S\Theta=0[/tex] but i don't see how . Can anyone help?
 
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  • #2
The spin of a tensor is determined by the transformation properties of its components under rotations. In two dimensions, rotations are generated by the angular momentum operator, so the spin of the components can be determined by looking at the action of the angular momentum operator on them. For example, since T^{zz} transforms as a tensor of rank two, it will have spin 2 under rotations. This can be seen by considering the action of the angular momentum operator on T^{zz}:L_zT^{zz}=i\partial_zT^{zz}-i\partial_{\bar z}T^{zz}This equation implies that T^{zz} transforms as a vector with spin 2 under rotations. Similarly, T^{\bar z \bar z} will have spin -2 under rotations, since it is a tensor of rank two and has the opposite transformation properties of T^{zz}. Finally, since the trace of a tensor is invariant under rotations, $\Theta$ will have spin 0.
 
  • #3


I would first clarify that the spin of a quantity refers to its representation under rotations in a given space. In this case, the space in question is two-dimensional, so the spin will be a two-dimensional representation.

To find the spin of the components T^{zz}, T^{\bar{z}\bar{z}}, and \Theta, we need to consider how they transform under rotations. In two dimensions, rotations are described by a single parameter, usually denoted by \phi.

We can write the rotation transformation for the complex coordinates z and \bar{z} as:

z \rightarrow z' = e^{i\phi} z
\bar{z} \rightarrow \bar{z}' = e^{-i\phi} \bar{z}

Using the transformation properties of the stress-energy tensor, we can then determine the spin of each component. The spin of a tensor is defined as the difference between the number of indices that transform under rotation and the number of indices that are invariant under rotation.

For T^{zz}, we see that it transforms under rotation as:

T^{zz} \rightarrow T^{zz'} = e^{2i\phi} T^{zz}

So, T^{zz} has two indices that transform under rotation, and no indices that are invariant. Therefore, the spin of T^{zz} is 2.

Similarly, for T^{\bar{z}\bar{z}}, we have:

T^{\bar{z}\bar{z}} \rightarrow T^{\bar{z}\bar{z}'} = e^{-2i\phi} T^{\bar{z}\bar{z}}

So, T^{\bar{z}\bar{z}} has two indices that transform under rotation, and no indices that are invariant. Therefore, the spin of T^{\bar{z}\bar{z}} is also 2.

For \Theta, we see that it transforms under rotation as:

\Theta \rightarrow \Theta' = \Theta

So, \Theta has no indices that transform under rotation, and two indices that are invariant. Therefore, the spin of \Theta is 0.

From these calculations, we can confirm that the authors of the paper are correct in their claim that the spin of T^{zz} is 2, the spin of T^{\bar{z}\bar{z}} is -2, and the spin of \
 

Related to Spin of the sress-energy tensor in d=2

1. What is the stress-energy tensor in d=2?

The stress-energy tensor is a mathematical quantity that describes the distribution of energy and momentum in a given space or system. In d=2, it specifically refers to a two-dimensional space or system.

2. How is the stress-energy tensor calculated in d=2?

The stress-energy tensor in d=2 is calculated using the Einstein field equations, which relate the curvature of space-time to the distribution of energy and momentum within it. This calculation involves taking into account the contributions of matter, radiation, and other sources of energy and momentum.

3. What is the significance of the spin of the stress-energy tensor in d=2?

The spin of the stress-energy tensor in d=2 refers to the intrinsic angular momentum of the tensor itself. It is a fundamental property of the tensor that can affect how it interacts with other physical quantities, such as gravitational fields.

4. Can the spin of the stress-energy tensor in d=2 be measured?

No, the spin of the stress-energy tensor in d=2 cannot be directly measured. It is a mathematical abstraction that is used to describe physical phenomena and make predictions about their behavior.

5. How does the spin of the stress-energy tensor in d=2 relate to the concept of symmetry?

The spin of the stress-energy tensor in d=2 is related to the concept of symmetry through Noether's theorem, which states that for every continuous symmetry in a physical system, there exists a corresponding conserved quantity. In the case of the stress-energy tensor, its spin is related to the symmetry of space-time.

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