By Viatcheslav F. Mukhanov, Sergei Winitzki

This can be the 1st introductory textbook on quantum box concept in gravitational backgrounds meant for undergraduate and starting graduate scholars within the fields of theoretical astrophysics, cosmology, particle physics, and string conception. The e-book covers the elemental (but crucial) fabric of quantization of fields in an increasing universe and quantum fluctuations in inflationary spacetime. It additionally incorporates a special clarification of the Casimir, Unruh, and Hawking results, and introduces the tactic of potent motion used for calculating the back-reaction of quantum structures on a classical exterior gravitational box. The wide scope of the fabric coated will give you the reader with an intensive point of view of the topic. each significant result's derived from first ideas and carefully defined. The booklet is self-contained and assumes just a easy wisdom of basic relativity. routines with distinctive strategies are supplied through the ebook.

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N! 7) 35 3 Quantizing a driven harmonic oscillator √ (The factors n! ; in other words, all states of the oscillator are of the form |ψ = ∞ n=0 ψn |n , ∞ n=0 2 |ψn | < ∞. 8) Remark: why is {|n } a complete basis? A description of a quantum system must include not only the algebra of quantum operators but also a specification of a Hilbert space in which these operators act. 8) cannot be derived from the commutation relation [ˆ q , pˆ] = i without additional assumptions. In fact, if one assumes the existence of a unique normalized eigenvector |0 such that a ˆ− |0 = 0, as well as the diagonalizability of the Hamiltonian, then one can prove that the vectors {|n } form a complete basis in the Hilbert space.

For the free field, we set ˆ ≡ H 48 d3 k ωk a ˆ+ ˆ− ka k. 3 The Schrödinger equation for a quantum field After this redefinition, the vacuum state becomes an eigenstate of zero energy: ˆ |0 = 0. 0| H The resulting quantum theory agrees with experiments. 3 The Schrödinger equation for a quantum field So far we have been working in the Heisenberg picture, but fields can be quantized also in the Schrödinger picture. Here we first consider the Schrödinger equation for a collection of harmonic oscillators and then generalize that equation to quantum fields.

We found that the vacuum state |0in is expressed as the linear combination 1 2 |0in = exp − |J0 | 2 ∞ Jn √0 |nout , n! 11) 37 3 Quantizing a driven harmonic oscillator or equivalently 1 2 ˆ+ |0in = exp − |J0 | + J0 a out |0out . 2 This formula is similar to the definition of a coherent state of the harmonic oscillator. Indeed, one can verify that |0in is an eigenstate of a ˆ− out with eigenvalue J0 . 11) shows that the state describing the early-time vacuum is a super2 position of excited states at late times, having the probability |Λn | for the occupation number n.