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The commutation rules assume the form in VS .NET
The commutation rules assume the form PDF417 Recognizer In .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. PDF417 2d Barcode Encoder In VS .NET Using Barcode printer for VS .NET Control to generate, create PDF417 2d barcode image in .NET framework applications. [<J>(k), <J>(k')] = [ii(k), ii(k')] = 0 [<J>(k),ii(k')] = i2::e i(kk')n
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Generate Code 128 Code Set B In None Using Barcode encoder for Software Control to generate, create Code 128 Code Set A image in Software applications. Barcode Generator In VB.NET Using Barcode printer for VS .NET Control to generate, create bar code image in VS .NET applications. Wk = Wk = Jm 2 ak
 cos k) ...;4nwk
~ [wk<J>(k) + iii(k)] ~ [wk<J>t(k) iiit(k)] (316) at =
...;4nwk
[ak' ak'] = c5(k  k') The operators at and ak create or destroy the mode k with energy Wk' The hamiltonian H reads 1 f+1t f+1t (317) H ="2 1t dk wk(atak + akat) = 1t dk Hk while the fields are expressed as (318) 110 QUANTUM FIELD THEORY
The denomination of ak and al as destruction and creation operators is justified by the fact that if IE) is an eigenstate of H with energy E we have + [H, ak] IE) = (E  wk)ak IE) HallE) = EallE) + [H, IE) = (E + wk)alIE) Hak IE) Eak IE) (319) The mode k of energy Wk is interpreted in this mechanical model as a coherent quantized vibration of the lattice atoms, or phonon. Here we understand clearly the relation between particles (phonons) and fields (CPn is the displacement of the nth atom). Our physical intuition might lead us first to consider states of the crystal characterized by a wave function obtained by diagonalizing the field We shall, rather, choose to generate the states starting from the ground state 10) and its excitations in terms of phonons. We encounter at once a difficulty since the groundstate energy, the lowest eigenvalue of the hamiltonian, is in fact infinite. Each mode k contributes an amount iWk to the zero point energy. This is in agreement with the 'uncertainty relations since each oscillator has a minimum momentum spread due to its potential energy. Since we have a continuum of modes, each slice (k, k + dk) leads to an infinite energya fact to be traced to the infinite size of the system. Indeed, since <Olal = 0
(320) and the last integral is meaningless since [ak> al] = b(O)!. If, however, the crystal is of finite size ( N :::;; n :::;; N) let us take periodic boundary conditions by identifying the sites nand n + p(2N + 1), realizing a circular arrangement. The wave vector k would then be restricted to values k = [2n/(2N + l)]q with q an integer and  N :::;; q :::;; N, and the zero point energy (N /2N) I ~ ~ Wq would be finite. Clearly (1/2N) I~~Wq has a finite limit as N tends to infinity, showing that the above energy is indeed proportional to the size of the system. By taking a discrete instead of a continuous model, we have introduced a Brillouin zone n :::;; k :::;; n in momentum space, equivalent to an ultraviolet cutoff in the original model. The reader will recall that the origin of the expression "ultraviolet catastrophe" stems from the divergent contribution to the blackbody thermal radiation of highfrequency modes of the electromagnetic field. A space cutoff allows, further, an unambiguous definition of the hamiltonian operator. This is referred to by saying that we put the system in a box. We can, however, use the following device. Zero point energy is unobservable unless we destroy the crystal. In field theory, the ground state will be interpreted as the vacuum and it will be even harder to destroy! Energy exchanges with the crystal are insensitive to the choice of an origin. We declare by fiat that the ground state has zero QUANTIZATIONFREE FIELDS
energy and we redefine the hamiltonian as
(321) f+,," dk wkalak
We shall, of course, have to make sure in relativistic theories that this procedure preserves Lorentz covariance. In this new expression the creation and annihilation operators appear in "normal order," the latter to the right of the former. This is also called Wick's ordering and is denoted by a doubledot symbol: :i(alak + akal):

