Finite size effect

introduction

• Casimir effect in QED is one example of finite size effect
• the stress on the bounding surfaces when quantum field is confined to finite volume of space
• type of boundaries
  
  • real material media
  • interface between two different phases of the vacuum of a field theory such as QCD, in which case colored field may only exist in the interior region
  • topology of space
• the boundaries restrict the modes of the quantum fields 
• give rise to measurable and important forces

 

how to compute the Casimir effect

• zero-point energy in the presence of the boundaries
  
  • sum over all modes
  • any kind of constraint or boudary conditions on the the zero-point modes of the quantum fields in question, including backgrounds such as gravity
    
  • In a model without boundary conditions, the Hamiltonian value associated wih the vacuum or ground state, called zero-point energy, is usually discarded because, despite being infinite, may be reabsorbed in a suitable redefinition of the energy origin
  • there are several ways to put such an adjustment into practice, normal ordering being oneof the most popular
• Green's functions method
  
  • represents the vacuum expectation value of the produc of fields
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conformal transform from the plane to cylinder

• \(z \to w=\frac{L}{2\pi}\ln z\)
• energy momentum tensor changes
  \(T_{cyl}(w)=-(\frac{2\pi}{L})^2\{T_{pl}(z)z^2-\frac{c}{24}\}\)
  \(L_0 \to L_0-c/24\)
  
• the central charge emerges
• central charge is proportional to the Casimir energy, the change in the vacuum energy density brought about by the periodicity condition on the cylinder

 

 

 

free energy

\(F(L)=\frac{c\pi}{6L}\)

 

 

related items

• cosmological constant
  

 

 

표준적인 도서 및 추천도서

• 찾아볼 수학책
• The Casimir Effect: Physical Manifestations of Zero-Point Energy
  
  • [1]Kimball A. Milton
• Conformal invariance and finite size effects in critical two dimensional statistical models
  
  • Claude Itzykson
• Casimir effect in critical systems
  
  • Michael Krech
• http://gigapedia.info/1/Casimir+effect
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

참고할만한 자료

• http://ko.wikipedia.org/wiki/카시미르효과
• http://en.wikipedia.org/wiki/finite_size_effect
• http://en.wikipedia.org/wiki/Casimir_effect
• http://en.wikipedia.org/wiki/Vacuum_energy
• http://en.wikipedia.org/wiki/

 

 

articles

• Conformal invariance, the central charge, and universal finite-size amplitudes at criticality

 

 

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