By H. Dekker

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**Extra resources for Classical and Quantum Mechanics of the Damped Harmonic Oscillator Dekker**

**Example text**

2) the interaction term is strictly local in order not to violate relativistic microcausality. Canonically quantized, such local equations and their perturbation series are nonrenormalizable. To avoid divergences Heisenberg decided to give up canonical quantization. After some preliminary studies he introduced regularizing dipole ghosts, which in consequence led to the indefinite metric of the theory. , an approximate evaluation of the many-time Schwinger–Dyson–Freese equations of the theory. In the lowest orders of Tamm–Dancoff approximations some successful calculations of states, coupling constants and scattering processes in the pre-standard model physics were performed, see the book by Heisenberg [Heis 66], but the theory has serious drawbacks which did not admit a further continuation along these lines.

One of these tensors is3: This tensor is of Maxwell type because in electromagnetical terms, we have (for the electric photon): Except for the mass terms, we recognize the Maxwellian form. And we have: 3 The factor potentials. 4 1 The Theory of Light and Wave Mechanics The Photon Spin Let us calculate the angular momentum with the nonsymmetric tensor is not a constant of motion. But, as in Dirac’s theory, we can find a constant of motion if we add an appropriate term: with The dual of this tensor in is a pseudo-vector.

70): Here, the power of E is not but , so that we find a result opposite to the result for — If n is odd, has the same indefinite sign as energy: this was the case for for a Dirac electron. — If n is even, the sign of is positive definite as for the photon and as will be the case for the graviton. This is confirmed by more sophisticated calculations using the energy tensor density. We shall introduce two classes of tensors. 36), given by recipes of quantum mechanics. 40); it is inspired by electromagnetism.

### Classical and Quantum Mechanics of the Damped Harmonic Oscillator Dekker by H. Dekker

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