The canonical commutators

The previous discussion suggests that commutation relations that hold for operators in first quantization, do not necessarily hold for their second-quantization counterparts in a finite basis. Consider  

In these expressions, square brackets around a first-quantization operator represent the one-electron integral of this operator in the given basis. This somewhat cumbersome notation is adopted for this discussion to make the dependence of the integrals on the first-quantization operators explicit. In Section 1.8, the commutator between the two excitation operators is shown to be 

The second-quantization canonical commutator therefore becomes proportional to the number operator in the limit of a complete basis  

This expression should be compared with its first-quantization counterpart (1.5.20). For finite basis sets, the second-quantization canonical commutator turns into a general one-electron operator (1.5.28). 

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The second axiom, which is reminiscent of Mach s principle, also contains the seeds of Leibniz s Monads [reschQl]. All is process. That is to say, there is no thing in the universe. Things, objects, entities, are abstractions of what is relatively constant from a process of movement and transformation. They are like the shapes that children like to see in the clouds. The Einstein-Podolsky-Rosen correlations (see section 12.7.1) remind us that what we empirically accept as fundamental particles - electrons, atoms, molecules, etc. - actually never exist in total isolation. Moreover, recalling von Neumann s uniqueness theorem for canonical commutation relations (which asserts that for locally compact phase spaces all Hilbert-space representations of the canonical commutation relations are physically equivalent), we note that for systems with non-locally-compact phase spaces, the uniqueness theorem fails, and therefore there must be infinitely many physically inequivalent and... 

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

Here q and p are Heisenberg operators, y is the usual damping coefficient, and (t) is a random force, which is also an operator. Not only does one have to characterize the stochastic behavior of g(t), but also its commutation relations, in such a way that the canonical commutation relation [q(t), p(t)] = i is preserved at all times and the fluctuation-dissipation theorem is obeyed. ) Moreover it appears impossible to maintain the delta correlation in time in view of the fact that quantum theory necessarily cuts off the high frequencies. ) We conclude that no quantum Langevin equation can be obtained without invoking explicitly the equation of motion of the bath that causes the fluctuations.1 That is the reason why this type of equation has so much less practical use than its classical counterpart. 

The above development indicates that commutation relationships that hold for first quantization operators do not necessarily hold for second quantization operators in a finite one-electron basis. Consider the canonical commutators... 

For a complete one-electron basis, the canonical commutators become proportional to the number operators. For finite basis sets, the canonical commutator becomes a general one-body operator. 

This could be related to a commutation relation among the integral operators. Typical relations among the infinitesimal operators can be derived from this approach. He had come close to a derivation of the canonical commutation relation from the definition of the derivative of an operatorvalued function of a real variable. Before this canonical commutation, Bom considered the assumption of a complex domain of numbers ... 

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

The projected nature of the second-quantization operators has many ramifications. For exan le, relations that hold for exact operators such as the canonical commutation properties of the coordinate and momentum operators do not necessarily hold for projected operators. Similarly, the projected coordinate operator does not commute with the projected Coulomb repulsion operator. It should be emphasized, however, that these problems are not peculiar to second quantization but arise whenever a finite basis is employed. They also arise in first quantiztttion. but not until the matrix elements are evaluated. 

Hence the radiation operators A x), Av(y) do not satisfy the usual canonical commutation rules. Bather, the presence of such factors as d aj3].(/ 2) implies that their commutation rules are more singular... 

Hence, again these ip operators do not obey canonical commutation rules due to the presence of the factor J da2Pl(a2) (which is found to be divergent in perturbation expansion of the theory). 

To summarize the theory dynamic correlations are described by the unitary operator exp A acting on a suitable reference funchon, where A consists of excitation operators of the form (4). We employ a cumulant decomposition to evaluate all expressions in the energy and amphtude equations. Since we are applying the cumulant decomposition after the first commutator (the term linear in the amplimdes), we call this theory linearized canonical transformation theory, by analogy with the coupled-cluster usage of the term. The key features of the hnearized CT theory are summarized and compared with other theories in Table II. 

First consider a Hartree-Fock reference function and transform to the Fermi vacuum (aU occupied orbitals are in the vacuum). Then all particle density matrices are zero and the cumulant decomposition, Eq. (23), based on this reference corresponds to simply neglecting aU three and higher particle-rank operators generated by commutators. This type of operator truncation is used in the canonical diagonalization theory of White [22]. 

In addition to the encouraging numerical results, the canonical transformation theory has a number of appealing formal features. It is based on a unitary exponential and is therefore a Hermitian theory it is size-consistent and it has a cost comparable to that of single-reference coupled-cluster theory. Cumulants are used in two places in the theory to close the commutator expansion of the unitary exponential, and to decouple the complexity of the multireference wave-function from the treatment of dynamic correlation. 

Let us consider a system in equilibrium, described in the absence of external perturbations by a time-independent Hamiltonian Ho. We will be concerned with equilibrium average values which we will denote as (...), where the symbol (...) stands for Trp0... with p0 = e H"/ Vre the canonical density operator. Since we intend to discuss linear response functions and symmetrized equilibrium correlation functions genetically denoted as Xba(, 0 and CBA t,t ), we shall assume that the observables of interest A and B do not commute with Ho (were it the case, the response function %BA(t, t ) would indeed be zero). This hypothesis implies in particular that A and B are centered A) =0,... 

Theorem. Let G be an algebraic affine group scheme. Then 7c0(Jc[G]) represents an etale group n0 G, and all maps from G to etale groups factor through the canonical map G - jr0 G. The kernel G° of this map is a connected closed normal subgroup represented by the factor ofk[G] on which s is nonzero. The construction of ic0G and G° commutes with base extension. 

In terms of these results it can be shown that the value of a Poisson bracket is invariant under a canonical transformation of the coordinates. Like the corresponding commutator relationships in quantum mechanics, to which it is related by the expression... 

Generalized momentum operators as defined by Eq. (2.77) can be used in wave mechanical as well as in matrix mechanical formulations. It ensures that the operators are Hermitian, and that momenta, 7r, conjugated to generalized coordinates, qh fulfil commutation relations similar to the canonical relations of Cartesian coordinates and momenta,... 

Here, we call the attention of the reader that our Eq. (24) in the previous interlude correspond to Eqs. (32) in Ref. [3] for the cartesian components of the angular momentum in the body frame and the inertial frame, respectively, in terms of the Euler angles. Notice that the angles if and

commutation rules from Eq. (22) in Ref. [3], for the analysis of the rotations of asymmetric molecules are as follows ... 

The non-commutativity which presents itself here is not, however, of the most general kind, as the theory shows for the left-hand expression, with a pair of canonically conjugated variables, can take... 

We now introduce the ideas of Weyl to distinguish between pure states and mixtures. Pure states were mathematically represented by eigenvectors of observables, which described the properties of a particle or a dynamic state. On the other hand, mixtures were composed of pure states of a certain mixing relationship. These aspects are clearly important to chemists and obviously to the electrochemists too. The canonical variables, G and H [19], have to satisfy the canonical or Heisenberg commutation relation, derived from Equations 3.12 and 3.13 ... 

The non-commutative nature of the multiplication of matrices is of great importance in matrix mechanics. The difference of the product of the matrix q,- representing the coordinate gr, and the matrix p, representing the canonically conjugate momen-... 

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