Defining Spin in QFT in Curved Spacetime

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SUMMARY

This discussion focuses on defining spin in Quantum Field Theory (QFT) within curved spacetime. In flat spacetime, spin is invariant under the Poincaré group and is represented by SO(3) generators. However, in curved spacetime, the concept of particles becomes ambiguous, necessitating a shift towards fields. The proper approach involves forming a bundle over the spacetime with fibers analogous to those in Minkowski space, particularly in globally hyperbolic and stationary spacetimes. For further insights, refer to Mikio Nakahara's "Geometry, Topology and Physics, Second Edition," specifically Section 11.6, which elaborates on the definition and limitations of spinor fields.

PREREQUISITES
  • Understanding of Quantum Field Theory (QFT)
  • Familiarity with curved spacetime concepts
  • Knowledge of Poincaré and SO(3) groups
  • Basic grasp of fiber bundles in differential geometry
NEXT STEPS
  • Study the implications of spinor fields in curved spacetime
  • Explore the mathematical framework of fiber bundles in QFT
  • Investigate globally hyperbolic and stationary spacetimes
  • Read Mikio Nakahara's "Geometry, Topology and Physics, Second Edition" for detailed insights
USEFUL FOR

The discussion is beneficial for theoretical physicists, mathematicians specializing in differential geometry, and researchers focused on Quantum Field Theory in curved spacetime.

paweld
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How one can define a spin in Qunatum Filed Theory in curved spacetime. If the
space is flat it's invarainat under Poincare group - so in particular it's invariant under SO(3).
Spin operators are simply generators of SO(3). If the space isn't flat we cannot define
spin in this way. I know that in curved spacetime we should think of fileds rather than particles,
because notion of particle is not always well defined. So maybe better question is
how we define spin in globally hyberbolic and stationary spacetime?
 
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You just form a bundle over the spacetime with the same Fiber as in the Minkowski case.

See Mikio Nakahara "Geometry, Topology and Physics, Second Edition" Section 11.6

It not only tells you how to define it, but what sort of manifolds do not admit a spinor field.
 

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