Can Any Continuous Coordinate Transformation Be a Local Poincare Transformation?

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Discussion Overview

The discussion revolves around whether any continuous coordinate transformation on a differential manifold can be considered a local Poincaré transformation in every tangent space of that manifold. The scope includes theoretical considerations related to differential geometry and gauge theories.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant questions if all continuous coordinate transformations can be viewed as local Poincaré transformations.
  • Another participant argues that certain transformations, such as scaling (e.g., x → 2x), do not qualify as Poincaré transformations.
  • A subsequent reply suggests that excluding scale transformations may allow for a broader inclusion of continuous coordinate transformations as local Poincaré transformations.
  • One participant references gauge theories of the Poincaré group, implying a connection to the discussion but does not clarify how it relates to the original question.
  • Another participant asserts that the Poincaré group may not be well-defined on generic manifolds, particularly those with dimensional discrepancies.
  • Infinitesimal diffeomorphisms are introduced, with a focus on the condition of "no scalings" to ensure that certain transformations are Lorentz matrices.
  • A challenge is posed regarding the applicability of these concepts to one-dimensional manifolds, questioning the definition of a Poincaré transformation in that context.

Areas of Agreement / Disagreement

Participants express disagreement regarding the conditions under which continuous coordinate transformations can be classified as local Poincaré transformations. There is no consensus on the applicability of these transformations, particularly in relation to scaling and dimensionality.

Contextual Notes

There are unresolved assumptions regarding the definitions of Poincaré transformations and the specific characteristics of the manifolds under discussion. The implications of dimensionality on the applicability of the Poincaré group remain unclear.

micomaco86572
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Can any continuous coordinate transformation on a differential manifold be viewed as a poincare transformation locally in every tangent space of this manifold?

Thx!
 
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No. Say your manifold is the real line, and your coordinate transformation is x\rightarrow 2x. That's not a Poincare transformation.
 
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 said:
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.
 
Last edited by a moderator:
micomaco86572 said:
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).
 
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.
 
arkajad said:
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?
 
bcrowell said:
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!
 

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