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

 

 

•  

 

 

conformal transform from the plane to cylinder

• strip geometry

• transformaion
  \(z \to w=\frac{L}{2\pi}\ln z\)
  maps the entire plane onto a strip of width L
  
• Schwarzian derivative
  \(\frac{1}{2z^2}\)
  
• 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

 

 

vacuum energy density

\(<T_{cyl}(w)>=-\frac{c\pi^2}{6L^2}\)

 

 

free energy per unit length

• periodic boundary condition (i.e. infinitely long cylinder of circumference L)
  \(F_L=f_{0}L-\frac{c\pi}{6L}\)
  where \(f_{0}\) is the free energy per unit area in the thermodynamic \(L\to\infty\) limit
  

 

 

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=

 

 

encyclopedia

• 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

• Universal term in the free energy at a critical point and the conformal anomaly
  
  • Ian Affleck, Phys. Rev. Lett. 56, 746–748 (1986)
• Conformal invariance, the central charge, and universal finite-size amplitudes at criticality
  
  • H. W. J. Blöte, J. Cardy and M. P. Nightingale, Phys. Rev. Lett. 56, 742–745 (1986)

• http://www.ams.org/mathscinet
• [2]http://www.zentralblatt-math.org/zmath/en/
• [3]http://arxiv.org/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://dx.doi.org/10.1103/PhysRevLett.56.746

 

 

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