Commutation equal time

We postulate the equal time commutation rules for the field operators to be... 

The canonical commutation rules, which the equal-time operators satisfy, are... 

Hence, by virtue of the equation of motion (10-1), which >fi(x) obeys, and the equal time commutation rules (10-8), the Green function GA obeys the following equation... 

The equal time commutations rules obeyed by ip and will be invariant provided that... 

The choice of ijr 2 — 1, together with the antiunitary character of U(it), guarantees the invariance of the equal time commutation rules under U(it). With these definitions of the transformation properties of the spin field operators one verifies that... 

Indeed, these operators satisfy the usual commutation relations at equal times... 

Here y5 = iyi Y2Y3Y4, and t6 are Pauli matrices. These define an algebra of equal time commutators ... 

In the first step one has to quantize the classical field theory. The standard canonical quantization via equal-time commutation relations for the fermion field operator % yields... 

What has already been said about space and time coordinates in the preceding chapters suggests the obvious question for which coordinates the Pauli principle is valid. Do we need to apply the pair permutation to only spatial coordinates or to space-time coordinates The permutation is to be applied to the spatial coordinates only since in quantum field theory the commutators are understood as equal-time commutation relations. Moreover, in nonrela-tivistic quantum mechanics this problem does not show up and we will later refer to space-spin coordinates that need to be exchanged for pair permutation. The situation will become more clear in section 8.6.5 once we have introduced the theoretical tools and background needed. 

The equal time commutation relations (20.2.5,6) then lead to... 

R = (i/ r) require translations t in addition to rotations j/. The irreducible representations for all Abelian groups have a phase factor c, consistent with the requirement that all h symmetry elements of the symmetry group commute. These symmetry elements of the Abelian group are obtained by multiplication of the symmetry element./ = (i/ lr) by itself an appropriate number of times, since R = E, where E is the identity element, and h is the number of elements in the Abelian group. We note that N, the number of hexagons in the ID unit cell of the nanotube, is not always equal h, particularly when d 1 and dfi d. 

Since div ( ) mid div 3 (x) commute with 8(x ) and 3 t (x ) for x0 —x, they have vanishing commutators with the hamiltonian and hence, they are time-independent operators. In fact, their constancy in tame implies that they commute with 3 (x) and S(x) at all times and hence they must be c-number multiples of the unit operator. If these c-numbers are set equal to zero initially, they will remain zero for all times. With this initial choice for div 8(x) and div 3tf(x), the operators S and satisfy all of the Maxwell equations (these now are operator equations ) ... 

Summarizing, we have noted that the Heisenberg operators Q+(t) obey field free equations i.e., that their time derivatives are given by the commutator of the operator with Ha+(t) = Ho+(0) and that this operator H0+(t) is equal to H(t) = H(0). The eigenstates of H0+ are, therefore, just the eigenstates of H. We can, therefore, identify the states Tn>+ with the previously defined >ln and the operator [Pg.602]

The second equality in eq. (16) follows because zp C Z, = Z(g, G), which, from the definition of the centralizer, all commute with g . The third equality follows because the double sum consists of the same / terms repeated z times as p runs from 1 to z. It follows... 

The second equality in eq. (11) follows because f>, E and the identity, which can only arise when l = k, is repeated ck times. Because the Dirac characters commute (eq. (4)), the triple product in eq. (11) is invariant under any permutation of i,j, k, so that... 

The first equality follows from the stationary character of the phase distribution,110 the second because the forming of averages is commutative. The third equality, however, is based on the statement that all motions of the set in question give the same time average for ( p). 

Thus, the element in row i and column k of the product matrix is the sum over j of the products (element j in row i of the first matrix) times (element j in column k of the second matrix). When AB equals BA. we say that the multiplication is commutative this property is limited to special pairs of square matrices of equal order. 

We shall use the principle of stationary action to obtain a variational definition of the force acting on an atom in a molecule. This derivation will illustrate the important point that the definition of an atomic property follows directly from the atomic statement of stationary action. To obtain Ehrenfest s second relationship as given in eqn (5.24) for the general time-dependent case, the operator G in eqn (6.3) and hence in eqn (6.2) is set equal to pi, the momentum operator of the electron whose coordinates are integrated over the basin of the subsystem 1. The Hamiltonian in the commutator is taken to be the many-electroii, fixed-nucleus Hamiltonian... 

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