By Ioan Merches
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18). 15) and relation µi µi = 1 have been used. 24), we observe that always ci ci = 0, because ni can be either 0 or 1; that is, we shall always have ni (1 − ni ) = 0. 19) is also satisfied. 3. Commutation relations and the Bose-Einstein statistics As seen in the previous paragraph, in the case of a field whose quanta obey the Bose-Einstein statistics the quantities ci and c+ i are operators and satisfy certain commutation rules. In the following investigation we shall reverse the way of our reasoning: show that the commutations relations lead - without any supplementary hypotheses - to the Bose-Einstein quantum statistics.
We also abandon the notation for the operators, and the symbols < |, | > which designate the bra- and ket-vectors, respectively. 5), we then have ∂ ∂ ∂ ∂ ∂ ∂ − =− = − ∂i ∂i = − . 4) to the state function ψ: p2 = pi pi = − ( − m2 )ψ = 0. 7) Since ψ is a scalar function, its Hermitian conjugate ψ + satisfies the same equation ( − m2 )ψ + = 0. ) of the field, it is necessary to construct the Lagrangian density first. 9) The Lagrangian formalism where ψ,i = ∂ψ . 7). 7). 8) is obtained by means of ∂L ∂ ∂L − ∂ψ ∂xi ∂ψ,i = 0.
Using the general framework developed in Chapter I, we are now able to write the fundamental quantities associated with the complex 51 The quantization of the scalar field scalar field (energy, momentum, and angular momentum). The expressions of these quantities for the real scalar field will then follow without any difficulty. Momentum four-vector. First of all, we notice that the tensor (c) Tik is already symmetric (there is no need to be symmetrized), while the scalar field quanta are spinless.