Field commutation relations

This operator can now be shown to be identical with the operator for an infinitesimal rotation of the vector field multiplied by i, i.e. J = — M. The components of the angular momentum operator satisfy the commutation relations... 

The pair operators do not satisfy simple commutation relations. Using the commutation relation for the Fermi field operators if/+(x), ip(x) we find the following relation for Cab(xxx2) ... 

When quantizing the theory, the commutation relations for the complex field u become... 

Finally, we make a few additional remarks. First, note that a pure number state is a3j= state whose phase 0k is evenly distributed between 0 and 2n. This is a consequence of the commutation relation [3] between Nk and e,0 <. Nevertheless, dipole mafKi w elements calculated between number states are (as all quantum mechanical amplitudes) well-defined complex numbers, and as such they have well-defined phajje j S Thus, the phases of the dipole matrix elements in conjunction with the mode ph f i f/)k [Eq. (12.15)] yield well-defined matter + radiation phases that determine the outcome of the photodissociation process. As in the weak-field domain, if only gJ one incident radiation mode exists then the phase cancels out in the rate expres4<3 [Eq. (12.35)], provided that the RWA [Eqs. (12.44) and (12.45)] is adoptedf However, in complete analogy with the treatment of weak-field control, if we irradh ate the material system with two or more radiation modes then the relative pb between them may have a pronounced effect on the fully interacting state, phase control is possible. 

Linear terms are absent because of the Brillouin theorem. The coefficients Ap p. and Bap p, can be calculated by equating the nonzero matrix elements of the RPA Hamiltonian [Eq. (122)], in the basis of Eq. (121), to the corresponding matrix elements of the exact Hamiltonian [Eq. (23)] in the same basis. From the translational symmetry of the mean field states it follows that the A and B coefficients do not depend on the complete labels P = n, i, K and P = n, /, K1, but only on the sublattice labels /, AT and /, K. The second ingredient of the RPA formalism is that we assume boson commutation relations for the excitation and de-excitation operators (Raich and Etters, 1968 Dunmore, 1972). 

In order to rigorously describe the nonlinear interaction between the weak pulsed fields, we now turn to the fully quantum treatment of the system. The traveling-wave electric fields can be expressed through single mode operators as j(z, t) = dj(t) Cqz (j = 1, 2), where uj is the annihilation operator for the field mode with the wavevector kp + q. The singlemode operators a and aq possess the standard bosonic commutation relations... 

The finite quantization bandwidth 5q for the field operators leads to the equaltime commutation relations... 

Therefore, whenever the normal form of the quadrature variance is negative, this component of the field is squeezed or, in other words, the quantum noise in this component is reduced below the vacuum level. For classical fields, there is no unity coming from the boson commutation relation, and the normal form of the quadrature component represents true variance of the classical stochastic variable, which must be positive. 

The quantum counterpart of the polarization matrices can be constructed in direct analogy to the field quantization [90]. We have to subject the field amplitudes in (124), (125), and (129) to the Weyl-Heisenberg commutation relations (22) and (23) respectively. Thus, we get the operator matrices of polarization of the multipole radiation of the form... 

We consider the input field operators at z = 0 for the mode of the frequency ooi and the input ones at z = 0 or z = L for the mode of the frequency 0)2 according to whether the propagation is co- or contradirectional, respectively. Further we consider the output field operators at z = L for the mode of the frequency ooi and the output ones at z Lor z = 0 for the mode of the frequency 0)2 according to the direction of propagation. We assume the ordinary commutation relations in cases k2<0... 

The basic expression for the quantization of the electromagnetic field is the expansion Eq(54). In the quantized theory the numbers Ck,, C x become operators of the creation C x and the annihilation Ck,x of photons. These operators are acting on the state vector < ) that is defined in the Fock space (occupation number space). The C xt Ck, operators satisfy the commutation relations ... 

Now the operators (a ), (x) may be called the operators of the quantized electron-positron field. These operators are defined in the Fock space and act on the state vector ). The creation and annihilation operators satisfy the anti-commutation relations ... 

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. 

Finally, we should note that all that has been said so far is valid for fermionic annihilation and creation operators only. In the case of bosons these operators need to fulfill commutation relations instead of the anticommutation relations. The fulfillment of anticommutation and commutation relations corresponds to Fermi-Dirac and Bose-Einstein statistics, respectively, valid for the corresponding particles. Accordingly, there exists a well-established cormection between statistics and spin properties of particles. It can be shown [65], for instance, that Dirac spinor fields fulfill anticommutation relations after having been quantized (actually, this result is the basis for the antisymmetrization simply postulated in section 8.5). Hence, in occupation number representation each state can only be occupied by one fermion because attempting to create a second fermion in state i, which has already been occupied, gives zero if anticommutation symmetry holds. 

This makes Maxwell s equations look simple. In quantum field theory things are more complicated because equations like (2.4.1), if interpreted literally as equations for the field operators, sometimes contradict the fundamental commutation relations that the fields must satisfy. Another common choice in QED is to have... 

When many-body interactions are weak, l f,o(N — l,j) Fi(N — l,j) and the states i f s (N - l,j) have essentially no spectral weight for s > 0, that is, the (N — 1) electron state is close to the frozen orbital state. The many-body matrix element then reduces to the one-electron matrix element mj [ < f(cf,lr) A(t) Pj < j(cj)> with the vector potential A(f) = Aoexp(—2 rivt) from the harmonic long wavelength radiation field (dipole approximation). The time-dependent factors in the wave functions and vector potential produce, after integration, the factor 5(cf — ej — hv), which is the one-electron approximation to the 5-function already anticipated in Eq. (3.2.2.4) and which reflects the conservation of energy. We have then for the matrix element (p(( (,k) Ao Pj j(cj)) where all the time-dependent factors have been removed. Owing to commutation relations, the operator Ao pj can be replaced by the operator A0 rj [19]. The practical form ofthe dipole matrix element for the emission from localized core levels is then... 

In the following, we will present two condensed versions of the episodes witnessed during the field visits related to the Daily Commute project. These episodes illustrate how understandings are re-produced in local scenes of activities, how different practices meet in the everyday, and how they woik together more or less... 

The field quantization can be performed by regarding the amplitudes a and oj as operators ajjg and, which satisfy the (canonical) commutation relations... 

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