"Finite size effect"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
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* Casimir effect in [[QED]] is one example of finite size effect | * 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 | * the stress on the bounding surfaces when quantum field is confined to finite volume of space | ||
| 10번째 줄: | 9번째 줄: | ||
* give rise to measurable and important forces | * give rise to measurable and important forces | ||
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==how to compute the Casimir effect== | ==how to compute the Casimir effect== | ||
| − | * zero-point energy in the presence of the boundaries | + | * zero-point energy in the presence of the boundaries |
** sum over all modes | ** 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 | + | ** 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 | ** 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 | ** there are several ways to put such an adjustment into practice, normal ordering being oneof the most popular | ||
| − | * Green's functions method | + | * Green's functions method |
** represents the vacuum expectation value of the product of fields | ** represents the vacuum expectation value of the product of fields | ||
| + | |||
| + | |||
| + | ==finite size effect and central charge== | ||
| + | * mass gap of order $1/N$ is the characteristic of conformal invariance | ||
| + | * finite-size correction term to the ground state energy | ||
| + | $$ | ||
| + | E_0=N\epsilon_0-\frac{\pi c v_F}{6N} +O(\frac{1}{N^2} | ||
| + | $$ | ||
| + | where $N$ denotes the number of sites in the spin chain | ||
| + | * finite-size corrections to largest eigenvalue of the transfer matrix | ||
| + | * low temperature asymptotics of free energy of quantum system | ||
| + | $$ | ||
| + | F(\beta)=F_0-\frac{\pi c}{6v_F}\beta^{-2}+O(\beta^{-2}) | ||
| + | $$ | ||
| + | where $\beta=T^{-1}$ is the inverse temperature | ||
| + | |||
==QFT interpretation of the Casimir effect== | ==QFT interpretation of the Casimir effect== | ||
| 63번째 줄: | 76번째 줄: | ||
* Ian Affleck [http://dx.doi.org/10.1103/PhysRevLett.56.746 Universal term in the free energy at a critical point and the conformal anomaly], Phys. Rev. Lett. 56, 746–748 (1986) | * Ian Affleck [http://dx.doi.org/10.1103/PhysRevLett.56.746 Universal term in the free energy at a critical point and the conformal anomaly], Phys. Rev. Lett. 56, 746–748 (1986) | ||
| − | + | * H. W. J. Blöte, J. Cardy and M. P. Nightingale [http://dx.doi.org/10.1103/PhysRevLett.56.742 Conformal invariance, the central charge, and universal finite-size amplitudes at criticality], Phys. Rev. Lett. 56, 742–745 (1986) | |
| + | * Cardy, John L. 1986. “Operator Content of Two-dimensional Conformally Invariant Theories.” Nuclear Physics. B 270 (2): 186–204. doi:http://dx.doi.org/10.1016/0550-3213(86)90552-3. | ||
2013년 8월 16일 (금) 16:29 판
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 product of fields
finite size effect and central charge
- mass gap of order $1/N$ is the characteristic of conformal invariance
- finite-size correction term to the ground state energy
$$ E_0=N\epsilon_0-\frac{\pi c v_F}{6N} +O(\frac{1}{N^2} $$ where $N$ denotes the number of sites in the spin chain
- finite-size corrections to largest eigenvalue of the transfer matrix
- low temperature asymptotics of free energy of quantum system
$$ F(\beta)=F_0-\frac{\pi c}{6v_F}\beta^{-2}+O(\beta^{-2}) $$ where $\beta=T^{-1}$ is the inverse temperature
QFT interpretation of the Casimir effect
books
- Kimball A. Milton The Casimir Effect: Physical Manifestations of Zero-Point Energy
- Claude Itzykson Conformal invariance and finite size effects in critical two dimensional statistical models
- Michael Krech Casimir effect in critical systems
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
articles
- Ian Affleck Universal term in the free energy at a critical point and the conformal anomaly, Phys. Rev. Lett. 56, 746–748 (1986)
- H. W. J. Blöte, J. Cardy and M. P. Nightingale Conformal invariance, the central charge, and universal finite-size amplitudes at criticality, Phys. Rev. Lett. 56, 742–745 (1986)
- Cardy, John L. 1986. “Operator Content of Two-dimensional Conformally Invariant Theories.” Nuclear Physics. B 270 (2): 186–204. doi:http://dx.doi.org/10.1016/0550-3213(86)90552-3.