By Leonard Susskind

Over the past decade the physics of black holes has been revolutionized by way of advancements that grew out of Jacob Bekenstein s consciousness that black holes have entropy. Stephen Hawking raised profound matters about the lack of details in black gap evaporation and the consistency of quantum mechanics in an international with gravity. for 2 a long time those questions questioned theoretical physicists and at last resulted in a revolution within the means we predict approximately house, time, topic and knowledge. This revolution has culminated in a notable precept referred to as The Holographic precept , that is now an incredible concentration of consciousness in gravitational examine, quantum box concept and simple particle physics. Leonard Susskind, one of many co-inventors of the Holographic precept in addition to one of many founders of String concept, develops and explains those techniques.

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Thus kX = 2nX π B where B is the size of the torus. 15) Entropy of the Free Quantum Field in Rindler Space 47 The entropy stored in the ﬁeld χk can be estimated from the entropy density of a 1+1 dimensional massless free boson at temperature T . 16) where T is the temperature. 18, the integral must be cut oﬀ when k > 1 . This is because when k = 1 the potential is already large at u = uo so that the entire contribution of χk is supressed. 19 we see two important features of the entropy of Rindler space.

However, unlike the case of a coherent superposition of states, the relative phases between the states |j are random. There is one special case when the density matrix is indistinguishable from a pure state. This is the case in which only one eigenvalue ρj is Quantum Fields in Rindler Space 35 nonzero. 12) A quantitative measure of the departure from a pure state is provided by the Von Neumann entropy S = −T r ρ log ρ = − j ρj log ρj . 13) S is zero if and only if all the eigenvalues but one are zero.

Finally, the Planck temperature is reached where totally new phenomena of an as yet unimagined kind take place. All of these phenomena have their place in the Fido’s description of the region near a horizon. In this lecture we will consider an enormously oversimpliﬁed description of the world in which only a single free ﬁeld is present in a ﬁxed space-time background. There is serious danger in extrapolating far reaching conclusions from so oversimpliﬁed a situation. In fact, the paradoxes and contradictions associated with black holes, quantum mechanics, and statistical thermodynamics that these lectures are concerned with are largely a consequence of such unjustiﬁed extrapolation.