Canonical commutators

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... 

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). 

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 ... 

Recall that S is a Noetherian scheme of finite dimension. Let z Z —> S be a closed embedding and j U —> S be the complimentary open embedding. For any simplicial sheaf we have a canonical commutative square in the simplicial homotopy category of the form... 

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

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). 

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. 

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