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Nk = −µk . 22) Since k > i, the operator ∂n∂ k does not act on µi . 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.
0. 1) is satisfied by the eigenvalue α = 0. The eigenvalues taken by α are then α = 0, 1, 2, 3, ..... showing that α can be any natural number, including zero. This fact straightforwardly suggests that α is a number of particles, α = n. The value α = 0 corresponds to the vacuum state. 14) c+ |n >= n + 1 |n + 1 > . 14). Indeed, √ √ (c c+ − c+ c)|n >= n + 1 c |n + 1 > − nc+ |n − 1 > √ √ = ( n + 1)2 |n > −( n)2 |n >= |n > . 1), among other things, leads to the definition of the vector ket |n >, which describes the state of a system of n particles.
6) General problems of field quantization where k1 and k2 are two constants having value one, but whose dimensions are different from one. The corresponding dimension equation in LSV is S 2 T −2 V 2 = [k1 ]S 2 L−2 + [k2 ]S 2 L−2 V −2 , meaning that [k1 ] = V 2 ; [K2 ] = V 4 . But the only quantity in LSV with value one and dimension V is c, therefore the energy-momentum relation reads E 2 = p2 c2 + m2 c4 . 7) 2. In LSV , the Klein-Gordon-Schr¨odinger equation writes ( where − m2 )|ψ >= 0, is the D’Alembertian operator ∂ ∂ .