New PDF release: An Introduction to Field Quantization

By Y. Takahashi

ISBN-10: 0080128246

ISBN-13: 9780080128245

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Extra resources for An Introduction to Field Quantization

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This formalism proves to be more convenient than the usual Hamiltonian approach for certain types of problems such as, for example, the Green function method for lattice vibrations. We denote the primitive lattice vectors by al9 a2 and a3, and the positions of the unit cells by xi = 11*1 + ha2 + /3Ä3 with /, = 0, ± 1 , ±2, ± 3 , . . 2) where the Νέ are chosen to be even for the sake of convenience. , r) and the vector displacement of the κίΐι atom in the unit cell at the point Xi is denoted by φ^\χΐ91), or simply by φ^\χ, t) with a = 1, 2, 3.

39) is a consequence of the identity e(k0) d(k*) = 2 ^ - {Ö(ko-a>(k)) - d(k0+œ(k))}. 38). Thus, H =ijdx{2(etA(x)y-A(x)-v2A(x)v v2A(x)-A(x)} = ijdx l(d,A(x)Y + | (ο,Λ(χ))4 = \\dx v 26 {(e,A(x)f + (V X A (JC)) 2 }. 45) giving -I"M(JC) = [^(JC), P,]. 45) we have assumed that A(x) vanishes sufficiently rapidly as | x | tends to infinity. The commutator for A{x) can now be obtained: [Aix)9 Aj(x')] = / (du - -L· θβλ D(x - χ'). 39). 13). 50) 27 [Ch. II An Introduction to Field Quantization What is wrong with this solution?

The order of field quantities cannot therefore be changed arbitrarily. The well-known convention [A,B] = AB-BA, {A,B} = AB + BA, will be employed. The following symbols are used to denote various operations : + hermitian conjugation. * complex conjugation. / transposition for matrices (not for quantized operator). T transposition for quantized operator (not for matrices such as Dirac's gammas). x hermitian conjugation for quantized operator and complex conjugation for matrices. Hence, * ί τ = Xf = +.