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Limiting the Business Method Patent: A Comparison and Proposed Alignment of European, Japanese and United States Patent Law
That ideas should freely spread from one to another over the globe, for the moral and mutual instruction of man, and improvement of his condition…
Bose condensation and branes
When the cosmological constant is considered to be a thermodynamical variable
in black hole thermodynamics, analogous to a pressure, its conjugate variable
can be thought of as a thermodynamic volume for the black hole. In the AdS/CFT
correspondence this interpretation cannot be applied to the CFT on the boundary
but, from the point of view of the boundary gauge theory, varying the
cosmological constant in the bulk is equivalent to varying the number of colors
in the gauge theory. This interpretation is examined in the case of
, for SUSY Yang-Mills at large , and the
variable thermodynamically conjugate to , a chemical potential for color, is
determined. It is shown that the chemical potential in the high temperature
phase of the Yang-Mills theory is negative and decreases as temperature
increases, as expected. For spherical black holes in the bulk the chemical
potential approaches zero as the temperature is lowered below the Hawking-Page
temperature and changes sign at a temperature that is within one part in a
thousand of the temperature at which the heat capacity diverges.Comment: 9 pages, 1 figur
The intrinsic curvature of thermodynamic potentials for black holes with critical points
The geometry of thermodynamic state space is studied for asymptotically
anti-de Sitter black holes in D-dimensional space times. Convexity of
thermodynamic potentials and the analytic structure of the response functions
is analysed. The thermodynamic potentials can be used to define a metric on the
space of thermodynamic variables and two commonly used such metrics are the
Weinhold metric, derived from the internal energy, and the Ruppeiner metric,
derived from the entropy. The intrinsic curvature of these metrics is
calculated for charged and for rotating black holes and it is shown that the
curvature diverges when heat capacities diverge but, contrary to general
expectations, the singularities in the Ricci scalars do not reflect the
critical behaviour.
When a cosmological constant is included as a state space variable it can be
interpreted as a pressure and the thermodynamically conjugate variable as a
thermodynamic volume. The geometry of the resulting extended thermodynamic
state space is also studied, in the context of rotating black holes, and there
are curvature singularities when the heat capacity at constant angular velocity
diverges and when the black hole is incompressible. Again the critical
behaviour is not visible in the singularities of the thermodynamic Ricci
scalar.Comment: 35 pages, 3 figure
Symplectic geometry and Hamiltonian flow of the renormalisation group equation
It is argued that renormalisation group flow can be interpreted as being a
Hamiltonian vector flow on a phase space which consists of the couplings of the
theory and their conjugate \lq\lq momenta", which are the vacuum expectation
values of the corresponding composite operators. The Hamiltonian is linear in
the conjugate variables and can be identified with the vacuum expectation value
of the trace of the energy-momentum operator. For theories with massive
couplings the identity operator plays a central role and its associated
coupling gives rise to a potential in the flow equations. The evolution of any
quantity , such as -point Green functions, under renormalisation group flow
can be obtained from its Poisson bracket with the Hamiltonian. Ward identities
can be represented as constants of the motion which act as symmetry generators
on the phase space via the Poisson bracket structure.Comment: 30 page
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