We provide a derivation of holographic entanglement entropy for spherical entangling surfaces. Our construction relies on conformally mapping the boundary CFT to a hyperbolic geometry and observing that the vacuum state is mapped to a thermal state in the latter geometry. Hence the conformal transformation maps the entanglement entropy to the thermodynamic entropy of this thermal state. The AdS/CFT dictionary allows us to calculate this thermodynamic entropy as the horizon entropy of a certain topological black hole. In even dimensions, we also demonstrate that the universal contribution to the entanglement entropy is given by A-type trace anomaly for any CFT, without reference to holography.

Towards a derivation of holographic entanglement entropy

In decision theory, mathematical analysis shows that once the sampling distribution, loss function, and sample are specified, the only remaining basis for a choice among different admissible decisions lies in the prior probabilities. Therefore, the logical foundations of decision theory cannot be put in fully satisfactory form until the old problem of arbitrariness (sometimes called "subjectiveness") in assigning prior probabilities is resolved. The principle of maximum entropy represents one step in this direction. Its use is illustrated, and a correspondence property between maximum-entropy probabilities and frequencies is demonstrated. The consistency of this principle with the principles of conventional "direct probability" analysis is illustrated by showing that many known results may be derived by either method. However, an ambiguity remains in setting up a prior on a continuous parameter space because the results lack invariance under a change of parameters; thus a further principle is needed. It is shown that in many problems, including some of the most important in practice, this ambiguity can be removed by applying methods of group theoretical reasoning which have long been used in theoretical physics. By finding the group of transformations on the parameter space which convert the problem into an equivalent one, a basic desideratum of consistency can be stated in the form of functional equations which impose conditions on, and in some cases fully determine, an "invariant measure" on the parameter space.

Prior Probabilities

Feynman amplitudes, regarded as functions of masses, exhibit various singularities when masses of internal and external lines are allowed to go to zero. In this paper, properties of these mass singularities, which may be defined as pathological solutions of the Landau condition, are studied in detail. A general method is developed that enables us to determine the degree of divergence of unrenormalized Feynman amplitudes at such singularities. It is also applied to the determination of mass dependence of a total transition probability. It is found that, although partial transition probabilities may have divergences associated with the vanishing of masses of particles in the final state, they always cancel each other in the calculation of total probability. However, this cancellation is partially destroyed if the charge renormalization is performed in a conventional manner. This is related to the fact that interacting particles lose their identity when their masses vanish. A new description of state and a new approach to the problem of renormalization seem to be required for a consistent treatment of this limit.

Mass singularities of Feynman amplitudes

The Landau singularities of the amplitude calculated from an arbitrary Feynman graph are considered. It is shown that the discontinuity across a branch cut starting from any Landau singularity is obtained by replacing Feynman propagators by delta functions for those lines which appear in the Landau diagram. The general formula is a simple generalization of the unitarity condition. The discontinuity is then considered as an analytic function of the momenta and masses; it is shown that its singularities are a subclass of the singularities of the original amplitude which corresponds to Landau diagrams with additional lines. The general results are illustrated by application to some single loop graphs. In particular, the general formula gives an immediate calculation of the Mandelstam spectral function for fourth‐order scattering. Singularities not of the Landau type are discussed and illustrated by the third‐order vertex part.

Singularities and Discontinuities of Feynman Amplitudes

It has been $30\phantom{\rule{0.3em}{0ex}}\text{years}$ since the discovery of the Unruh effect. It has played a crucial role in our understanding that the particle content of a field theory is observer dependent. This effect is important in its own right and as a way to understand the phenomenon of particle emission from black holes and cosmological horizons. The Unruh effect is reviewed here with particular emphasis on its applications. A number of recent developments are also commented on and some controversies are discussed. Effort is also made to clarify what seem to be common misconceptions.

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The Unruh effect and its applications

A general quantum field theory is considered in which the fields are assumed to be operator‐valued tempered distributions The system of fields may include any number of boson fields and fermion fields A theorem which relates certain complex Lorentz transformations to the T C P transformation is stated and proved With reference to this theorem, duality conditions are considered, and it is shown that such conditions hold under various physically reasonable assumptions about the fields Extensions of the algebras of field operators are discussed with reference to the duality conditions Local internal symmetries are discussed, and it is shown that these commute with the Poincare group and with the T C P transformation

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On the duality condition for quantum fields

A general Hermitian scalar field, assumed to be an operator−valued tempered distribution, is considered. A theorem which relates certain complex Lorentz transformations to the TCP transformation is stated and proved. With reference to this theorem, duality conditions are considered, and it is shown that such conditions hold under various physically reasonable assumptions about the field. A theorem analogous to Borchers’ theorem on relatively local fields is stated and proved. Local internal symmetries are discussed, and it is shown that any such symmetry commutes with the Poincare

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On the duality condition for a Hermitian scalar field

Within the general framework of local quantum field theory a physically motivated condition on the energy-level density of well-localized states is proposed and discussed. It is shown that any model satisfying this condition obeys a strong form of the principle of causal (statistical) independence, which manifests itself in a specific algebraic structure of the local algebras (“split property”). It is also shown that the proposed condition holds in a free field theory.

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Causal independence and the energy-level density of states in local quantum field theory

The vertex operator is examined in lowest order perturbation theory. It is found that, as a function of the invariant momentum transfer ${q}^{2}={\mathrm{q}}^{2}\ensuremath{-}{{q}_{0}}^{2}$, it is analytic in a cut plane with the branch point on the negative real axis. A spectral representation (dispersion relation) may therefore be inferred. The threshold of the spectrum depends on the masses of all fields involved unless certain inequalities hold between the masses of the incident and outgoing particles on one hand and the particles in intermediate states on the other; in that case the threshold depends only on the intermediate masses.

Spectral Representations in Perturbation Theory. I. Vertex Function

The problem of how to associate a statistical ensemble of a quantum mechanical system with an incomplete set of measured ensemble averages is discussed. The view adhered to in this note is that the most chaotic ensemble consistent with the measured ensemble averages is a reasonable choice of ensemble. The properties of this ensemble are studied rigorously for the case when the quantum mechanical system is associated with a finite‐dimensional Hilbert space. The results are extended heuristically to some particular ensembles of many‐boson and many‐fermion systems.

Eyvind H. Wichmann

Papers

On the duality condition for quantum fields

On the duality condition for a Hermitian scalar field

Causal independence and the energy-level density of states in local quantum field theory

Spectral Representations in Perturbation Theory. I. Vertex Function

Density Matrices Arising from Incomplete Measurements