By Ioan Merches
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Extra info for Basics of quantum electrodynamics
1) Commutation relations and the Bose-Einstein statistics where, for convenience, we dropped the index i. 1) on the right by c and grouping the terms, we have (c+ c) c = c (c+ c − 1). 2) Suppose now that |α > is an eigenvector of the Hermitian operator c+ c, corresponding to the eigenvalue α, that is (c+ c) |α >= α |α > . 3), we have (c+ c)(c |α >) = c (c+ c − 1) |α >= (α − 1) (c |α >). 4) The last relation shows that, if |α > is a eigenvector of the operator c+ c corresponding to the eigenvalue α, then c |α > is also a eigenvector of the same operator, belonging to the eigenvalue (α − 1).
17) We have (c c+ ) = 1 0 0 0 0 0 0 1 (c+ c) = ; . 1) is satisfied: (c c+ ) + (c+ c) = 1 0 0 1 . 5. Alternative methods of field quantization In the previous sections of this chapter we justified the necessity of field quantization, by means of the expansion of the state function |ψ > in terms of the eigenfunctions |φi >. This way we ascertained that the coefficients ci intervening in the expansion of |ψ > are operators, that 43 General problems of field quantization obey a commutation rule for bosons, and an anti-commutation rule for fermions, depending on which category the state vector |ψ > stands for.
4) L′ (x′i ) = L(x′i − ξi ) ≈ L(x′i ) − ξi ′ , ∂xi where we neglected higher order terms. 5) where δL = L′ (xi ) − L(xi ). 7) or, if a series expansion is performed, δU(r) = − U(r),i ξi ; δU(r),l = − U(r),il ξi . 8), we then have: ∂L ξi = 0. 3). Next, we shall prove that both the existence of the angular momentum density Jikl and its conservation appear as a result of the invariance of L with respect to an infinitesimal rotation of the field. Consider the infinitesimal rotation x′i = xi + ωik xk ; δxi = ωik xk .
Basics of quantum electrodynamics by Ioan Merches