Introduction to Quantum Fields in Classical Backgrounds by Viatcheslav F. Mukhanov, Sergei Winitzki

By Viatcheslav F. Mukhanov, Sergei Winitzki

This can be the 1st introductory textbook on quantum box conception in gravitational backgrounds meant for undergraduate and starting graduate scholars within the fields of theoretical astrophysics, cosmology, particle physics, and string conception. The ebook covers the elemental (but crucial) fabric of quantization of fields in an increasing universe and quantum fluctuations in inflationary spacetime. It additionally features a distinctive clarification of the Casimir, Unruh, and Hawking results, and introduces the strategy of powerful motion used for calculating the back-reaction of quantum platforms on a classical exterior gravitational box. The large scope of the fabric lined will give you the reader with an intensive point of view of the topic. each significant result's derived from first rules and punctiliously defined. The ebook is self-contained and assumes just a uncomplicated wisdom of basic relativity. workouts with special suggestions are supplied in the course of the publication.

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Remark: fermions. A classical theory of fermionic fields can be built by considering spinor fields ψ µ (x) with values in an anticommutative (Grassmann) algebra, so that ψ µ ψ ν = −ψ ν ψ µ . The assumption of anticommutativity is necessary to obtain the correct anticommutation relations in the quantum theory. Consideration of fermionic fields is beyond the scope of this book. 1 Requirements for the action functional To choose an action for a field, we use the following guiding principles: 1. The action is real-valued and has an extremum.

14) In Eq. 14) we omitted the superscript (0) for brevity; below we shall always use the time-independent creation and annihilation operators and denote them by a ˆ± k. Remark: complex oscillators. The modes φk (t) are complex variables; each φk may be (1) (2) thought of as a pair of real-valued oscillators, φk = φk +iφk . Accordingly, the operators φˆk are not Hermitian and (φˆk )† = φˆ−k . In principle, one could rewrite the theory in terms of Hermitian variables, but it is mathematically more convenient to keep the complex modes φk .

The Green’s functions Gret (t, t′ ) and GF (t, t′ ) are defined by Eqs. 17). For t1,2 ≥ T , show that: (a) The expectation value of qˆ(t1 )ˆ q (t2 ) in the “in” state is 0in | qˆ (t1 ) qˆ (t2 ) |0in = 1 iω(t2 −t1 ) e + 2ω Z 0 T dt′1 Z T 0 ` ´ ` ´ ` ´ ` ´ dt′2 J t′1 J t′2 Gret t1 , t′1 Gret t2 , t′2 . (b) The in-out matrix element of qˆ(t1 )ˆ q (t2 ) is 0out | qˆ (t1 ) qˆ (t2 ) |0in 0out | 0in Z T Z T ` ´ ` ´ ` ´ ` ´ 1 iω(t2 −t1 ) e + dt′1 dt′2 J t′1 J t′2 GF t1 , t′1 GF t2 , t′2 . = 2ω 0 0 40 4 From harmonic oscillators to fields Summary: Collections of quantum oscillators.

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