World sheet EM tensor in complex coordinates

In summary: Notice that in the complex coordinates, the components of the energy-momentum tensor simplify to T_{zz}=T_{\bar{z}\bar{z}}=0 and T_{z\bar{z}}=T_{\bar{z}z}=0, therefore leaving only the diagonal components T_{zz} and T_{\bar{z}\bar{z}} as non-zero. This is why the book states the expression for the holomorphic component T_X(z)=T_{zz}. Does that make sense?In summary, the holomorphic component of the energy-momentum tensor in complex coordinates is given by T_X(z)=T_{zz}=-2 : \partial _z X \cdot \partial _z X
  • #1
da_willem
599
1
In my Book (Becker, Becker, Schwarz) it is stated (eq 3.23) that the holomorphic component of the EM tensor is given by

[tex]T_X(z)=T_{zz}=-2 : \partial _z X \cdot \partial _z X :[/tex]

Now why is the expression for the (holonorphic, zz, component of the) energy momentum tensor in complex coordinates?
 
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  • #2
da_willem said:
In my Book (Becker, Becker, Schwarz) it is stated (eq 3.23) that the holomorphic component of the EM tensor is given by

[tex]T_X(z)=T_{zz}=-2 : \partial _z X \cdot \partial _z X :[/tex]

Now why is the expression for the (holomorphic, zz, component of the) energy momentum tensor in complex coordinates?

I do not really understand the question, what is wrong with using complex coordinates?

nonplus
 
  • #3
See e.g. E. Kiritsis, hep-th/9709062, page 44.
Essentially, you introduce new coordinates z=x+iy and \bar{z}=x-iy and then transform the components of the energy-momentum to new coordinates by the usual rules of tensor transformation.
If your question is why complex coordinates are introduced at the first place, the answer is because the requirement of 2-dimensional conformal invariance is particularly easy to achieve with complex coordinates. See e.g. (6.1.6) and (6.1.7) in the reference above.
 
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  • #4
Hmm, they were so short about it in the book that I think they presumed the relation should be intuitively clear or common knowledge or something. Well, I tried the transformation to complex coordinates:

[tex] T(z)= T_{zz} = (\frac{\partial x^0}{\partial z})^2T_{00} + (\frac{\partial x^1}{\partial z})^2T_{11}[/tex]

As the off-diagonal elements are zero. Now using [itex]x^0=\frac{1}{2}(z+/bar{z})[/tex] and [itex]x^1=\frac{i}{2}(z-/bar{z})[/tex] we get that

[tex]T(z)=\frac{1}{4}(T_{00} -T_{11}[/tex]

which is zero, as the terms are equal. This is the correct result, the ws em tensor should vanish, but leaves me blank as to how they got that expression...
 
  • #5
da_willem said:
This is the correct result, the ws em tensor should vanish, but leaves me blank as to how they got that expression...
The same way you did. :smile:
 

1. What is the World Sheet EM Tensor in Complex Coordinates?

The World Sheet EM Tensor in Complex Coordinates is a mathematical representation of the electromagnetic field on a two-dimensional surface, known as the world sheet. It describes the strength and direction of the electric and magnetic fields at each point on the surface in terms of complex numbers.

2. Why is the World Sheet EM Tensor important in physics?

The World Sheet EM Tensor is important in physics because it allows us to understand and analyze the behavior of electromagnetic fields on curved surfaces, such as in string theory. It also helps us to make predictions and calculations about the interactions between particles and fields on these surfaces.

3. How is the World Sheet EM Tensor calculated?

The World Sheet EM Tensor is calculated by taking the derivative of the electromagnetic potential with respect to the complex coordinates of the world sheet. This results in a 2x2 matrix with complex elements, representing the electric and magnetic fields on the surface.

4. What is the relationship between the World Sheet EM Tensor and the Maxwell's equations?

The World Sheet EM Tensor is closely related to the Maxwell's equations, which are the fundamental equations governing electromagnetism. In particular, the components of the tensor correspond to the electric and magnetic field strengths in the Maxwell's equations, and the tensor itself satisfies the conservation laws of energy and momentum.

5. Are there any applications of the World Sheet EM Tensor?

Yes, the World Sheet EM Tensor has various applications in theoretical physics, particularly in string theory and other models that involve curved surfaces. It also has potential applications in understanding and predicting the behavior of electromagnetic fields in complex systems, such as in condensed matter physics and astrophysics.

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