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More generally, if F(kl, ... , k,) is a symmetric function then the normalized rparticle state is in .NET framework
More generally, if F(kl, ... , k,) is a symmetric function then the normalized rparticle state is Recognize PDF 417 In VS .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET applications. Generating PDF417 2d Barcode In VS .NET Using Barcode generator for VS .NET Control to generate, create PDF417 2d barcode image in Visual Studio .NET applications. (332) PDF 417 Recognizer In Visual Studio .NET Using Barcode recognizer for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications. Create Barcode In VS .NET Using Barcode creator for Visual Studio .NET Control to generate, create bar code image in Visual Studio .NET applications. It is an interesting exercise left to the reader to compute the wave function of a state with a fixed number of phonons, and in particular the ground state, in the basis which diagonalizes the field. Bar Code Recognizer In VS .NET Using Barcode recognizer for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications. Generating PDF417 2d Barcode In Visual C#.NET Using Barcode printer for .NET Control to generate, create PDF417 image in .NET framework applications. Canonical quantization has been performed at a fixed time, say t = O. However, the theory is invariant under time translation. We could therefore have chosen another reference time t and used the operators <f>n(t), nn(t) such that Create PDF417 2d Barcode In .NET Framework Using Barcode printer for ASP.NET Control to generate, create PDF 417 image in ASP.NET applications. PDF 417 Encoder In VB.NET Using Barcode printer for VS .NET Control to generate, create PDF417 2d barcode image in .NET framework applications. >n(t) = i[H, <f>n(t)] nn(t) = i[ H, nn(t)] Code 128C Maker In .NET Using Barcode drawer for Visual Studio .NET Control to generate, create Code 128C image in Visual Studio .NET applications. Painting Barcode In Visual Studio .NET Using Barcode maker for .NET framework Control to generate, create barcode image in .NET framework applications. nn(t) Print 2D Barcode In .NET Framework Using Barcode generation for .NET Control to generate, create Matrix Barcode image in .NET framework applications. Draw Identcode In .NET Framework Using Barcode creator for .NET framework Control to generate, create Identcode image in Visual Studio .NET applications. (333) UCC128 Printer In None Using Barcode generation for Software Control to generate, create EAN / UCC  13 image in Software applications. EAN13 Scanner In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. with
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UPC Symbol Recognizer In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. GS1 128 Encoder In None Using Barcode printer for Office Excel Control to generate, create GS1 128 image in Microsoft Excel applications. Clearly ak e iov, al eio>, satisfy canonical commutation rules, and Hand N are time independent. The relation between the observables at times t and 0 is a unitarily implemented canonical transformation. The matrix elements representing measurements are independent of the choice of the Heisenberg or Schrodinger picture <al eiHt A(O) e iHt la> a/ e iHt ) A(O)(e iHt la
(Schrodinger) (Heisenberg) = <al (e iHt A(O) e iHt ) la>
The structure ofEq. (334) clarifies the presence of positive and negative frequencies in the field. They correspond to the creation or destruction of phonon modes. The latter have honest positive energies (hw). Hence <p is not to be understood as a wave function. It is an operator in Fock space even though it is written in terms of the solutions to a wave equation. The superposition is weighted with operatorvalued amplitudes. This procedure is referred to as second quantization. The mechanical example of the crystal has prepared the way for the quantum treatment of the relativistic scalar field. The atomic interpretation of the vibrations will disappear but the analogy between phonons and particles remains. We shall have even fewer scruples in setting the vacuum energy equal to zero. 312 Scalar Field
For the quantized free scalar field with an hamiltonian given by Eq. (311) we have essentially to transcribe the previous formulas in threedimensional continuous space. The threemomenta (an interpretation to be confirmed below) 2 are denoted k, while k O also stands for Wk = Wk = + m 2 > O. Throughout this book, the phase space measure for bosons will be noted: d k d k 2 2 dk = (2n 2wk = (2n)4 2nb(k  m )(}(k ) (335) The last expression exhibits clearly the Lorentz invariance of this measure. This may be checked directly on d 3k/Wk: it is invariant under rotations; a boost of velocity tanh f) leaves invariant the transverse components kT as well as wl  ki. Therefore, dk~ =
(COSh f) (COSh f) + sinh f) ~~) dkL + sinh f) ~~)Wk
proving that dk~/Wk' = dkL!Wk' In terms of the creation and annihilation operators satisfying [a(k), at(k')] = (2n)32wkb3(k  k') [a(k), a(k')] [at(k), at(k')] (336) QUANTIZATIONFREE FIELDS
the conjugate fields qJ(x), n(x) at time t rules (3.3) are
qJ(x) n(x) 0, fulfilling the basic commutation
dk [a(k) eik x
+ at(k) e ik . x] (337) eik ' x at(k) eikox ] dk wk[a(k) The vacuum ground state is defined through
a(k) 10) = < 1 )=1 while the hamiltonian takes the form
2" dk Wk: at(k)a(k) + a(k)at(k): 1 dk wkat(k)a(k) (338) According to Chap. 1 it is natural to expect that the linear momentum operator P is given by
pi =
d3 x E)Oi(x) dk ki [at(k)a(k) ~ a(k)at(k)] = dk kjat(k)a(k) (339) Unlike the case of energy, no normal ordering is required. The modes k and  k compensate each other, so that the vacuum is translationally invariant, i.e., an eigenstate of P with zero eigenvalue. The operators H == po and pi commute, and [Pll, at(k)] = kllat(k) (340) showing that at(k) acting on a state adds a fourmomentum kll. Up to normal ordering the energy momentum tensor density is identical with its classical counterpart
(341) with a <p(x) replaced by n(x). At time t
= XO
the field qJ reads
qJ(t, x) = eiHt qJ(O, x) e iHt =
dk [a(k) e ik . x + at(k) eik . X] (342) This expression automatically fulfills the KleinGordon equation. If we use the +> symbol uov for
uov = u(ov)  (ou)v
we can write a(k) in terms of qJ(O, x) and n(O, x) as
a(k) = d 3 x e ik x [WkqJ(O, x) + in(O, x)] = d3 x [e ik ' x80qJ(t, x)]t=o
(343) QUANTUM FIELD THEORY
and the last expression is in fact time independent. Indeed, d 3 x eik x
80 <p(x) = d 3 X {e ik ' x 05<P(x) + [(m 2  L1) eik X] <p(x)} using the fact that k5 = m 2 + k 2 . In all applications we have to consider a normalized superposition of plane waves, in which case the integration by parts of the laplacian operator will be justified. Taking now into account the fact that <p(x) satisfies the KleinGordon equation, we find that the above expression vanishes. Hence

