"Finite size effect"의 두 판 사이의 차이

2010년 5월 6일 (목) 11:24 판

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

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

transformaion
\(z \to w=\frac{L}{2\pi}\ln z\)
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

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

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

@@ 35번째 줄: / 35번째 줄: @@
 <h5 style="line-height: 2em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">conformal transform from the plane to cylinder</h5>
-* <math>z \to w=\frac{L}{2\pi}\ln z</math>
+*   <br>
+*  transformaion<br><math>z \to w=\frac{L}{2\pi}\ln z</math><br>
+*  Schwarzian derivative<br><math>\frac{1}{2z^2}</math><br>
 *  energy momentum tensor changes<br><math>T_{cyl}(w)=-(\frac{2\pi}{L})^2\{T_{pl}(z)z^2-\frac{c}{24}\}</math><br><math>L_0 \to L_0-c/24</math><br>
 * the central charge emerges
@@ 44번째 줄: / 46번째 줄: @@
-<h5>vacuum e</h5>
+<h5>vacuum energy density</h5>
+<math><T_{cyl}(w)>=-\frac{c\pi^2}{6L^2}</math>
-<h5>free energy</h5>
+<h5>free energy per unit length</h5>
+*
 <math>F(L)=\frac{c\pi}{6L}</math>