Can Any Continuous Coordinate Transformation Be a Local Poincare Transformation?

[micomaco86572]

Can any continuous coordinate transformation on a differential manifold be viewed as a poincare transformation locally in every tangent space of this manifold?

Thx!

 

[bcrowell]

No. Say your manifold is the real line, and your coordinate transformation is $x\rightarrow 2x$. That's not a Poincare transformation.

 

[micomaco86572]

yeah, I see. But if we exclude this scale transformation， can we still say the coordinate transformation could be viewed as the local poincare transformation?

 

[arkajad]

This is what you do in gauge theories of the Poincare group.

http://www.springerlink.com/content/qv46n02uq4301315/" by K. Pilch.

 

[micomaco86572]

This is what you do in gauge theories of the Poincare group.

http://www.springerlink.com/content/qv46n02uq4301315/" by K. Pilch.

Yeah，so I am thinking about whether the local poincare transformation includes all the continuous coordinate transformations except the scale transformation mentioned above.

 

[bcrowell]

Yeah，so I am thinking about whether the local poincare transformation includes all the continuous coordinate transformations except the scale transformation mentioned above.

No. The Poincare group isn't even well defined on a generic manifold (e.g., one that has the wrong number of dimensions).

 

[arkajad]

Infinitesimally, we can expand a vector field (infinitesimal diffeomorphism) as:

$$\xi(x+h)^\mu =\xi^\mu (x)+a^\mu_\nu h^\nu +...$$

To kill the unwanted degrees of freedom we impose the condition of "no scalings" which amounts to assuming that $$a^\mu_\nu$$ is a Lorentz matrix.

 

[bcrowell]

Infinitesimally, we can expand a vector field (infinitesimal diffeomorphism) as:

$$\xi(x+h)^\mu =\xi^\mu (x)+a^\mu_\nu h^\nu +...$$

To kill the unwanted degrees of freedom we impose the condition of "no scalings" which amounts to assuming that $$a^\mu_\nu$$ is a Lorentz matrix.

Are you claiming that this works on a one-dimensional manifold? If not, then what conditions do you propose adding in addition to the ones given by the OP?

 

[arkajad]

Are you claiming that this works on a one-dimensional manifold?

First tell me your definition of a poincare transformation on a one-dimensional manifold. I am really curious!