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Extra resources for Basics of quantum electrodynamics
As we have seen, the spin Sikl is connected to the tensor rank of the field. 6), this gives gikl = 0, meaning (c) that Tik is always symmetric. 14), that the quanta of a scalar field have null spin, and vice-versa. One can then say that the tensor order of a field determines the spin of the quanta of that field. 3). Indeed, the quanta of the electromagnetic field, which is a first-order tensor field, are characterized by spin 1, while the gravitational (second-order tensor) field has spin 2, etc.
Consider the complete orthonormal set of vectors |n > and take it as the basis for the matrix representation of the operators c+ c, cc+ , c and c+ . : 0 0 0 1 (c+ c) = 0 0 . 0 0 ... 0 0 ... . 2 0 ... . 13), writes (c)n′ n =< n′ |c|n >= √ n < n′ |n − 1 >= As one can see, the elements of the matrix parallel to the principal diagonal: √ 0 1 √0 0 2 √0 0 0 (c) = 3 0 0 0 . . √ n δn′ ,n−1 . 16) (c) are arranged on a line 0 0 0 . ... . ... 13), the elements of (c+ ) are (c+ )n′ n =< n′ |c+ |n >= that is √ n + 1 < n′ |n + 1 >= 0 0 √0 0 1 √0 (c+ ) = 0 2 √0 0 0 3 .
Let us also pick two systems of observable quantities, associated with the quantum state of the particles. Consider two such quantities A and ˆ We also suppose that the B, with their associated operators Aˆ and B. e. the two systems of quantities 28 Necessity of field quantization can only be determined with some characteristic ’uncertainties’. 9) < χk |χk′ >= δkk′ . 12) depending on which system we want to determine, A or B. 13) ˆ is while the number of particles for which βk is a eigenvalue of B ′ d+ k dk = nk .