Canonical quantization

We teach our students that the infinite zero-point energy that arises when a free relativistic scalar (or Dirac-) field theory is canonically quantized can be subtracted (discarded) by a suitable redefinition of the energy-origin in other words, by normal ordering. However, the energy-origin can be re-defined only once, and only in homogenous space (i.e. without boundary conditions) and... 

X. Scalar Interferometry and Canonical Quantization from Whittaker s Potentials... 

The potential is again primary in canonical quantization, and it has been shown in Section IX that quantization of 0(3) electrodynamics does not lead to charged photons. 

X. SCALAR INTERFEROMETRY AND CANONICAL QUANTIZATION FROM WHITTAKER S POTENTIALS... 

Whittaker s early work [27,28] is the precursor [4] to twistor theory and is well developed. Whittaker showed that a scalar potential satisfying the Laplace and d Alembert equations is structured in the vacuum, and can be expanded in terms of plane waves. This means that in the vacuum, there are both propagating and standing waves, and electromagnetic waves are not necessarily transverse. In this section, a straightforward application of Whittaker s work is reviewed, leading to the feasibility of interferometry between scalar potentials in the vacuum, and to a trouble-free method of canonical quantization. 

Having derived the Proca equation in gauge-invariant form on the U(l) and 0(3) levels, canonical quantization can be attempted. Defining the photon mass in reduced units as... 

The locally gauge-invariant Lehnert field equation corresponding to Eq. (374) was derived as Eq. (350). The photon picks up mass from the vacuum itself, and having derived a locally gauge-invariant Proca equation, canonical quantization can be applied to produce a photon with mass with three space dimensions. 

The B cyclic theorem is a Lorentz invariant construct in the vacuum and is a relation between angular momentum generators [42], As such, it can be used as the starting point for a new type of quantization of electromagnetic radiation, based on quantization of angular momentum operators. This method shares none of the drawbacks of canonical quantization [46], and gives photon creation and annihilation operators self-consistently. It is seen from the B cyclic theorem ... 

The quantum Hamiltonian of the classical kicked rotor, defined by the classical Hamiltonian function (5.1.2), is easily obtained by canonical quantization. On replacing the classical angular momentum L by the quantum angular momentum operator L according to... 

The canonical quantization of the field has introduced by Dirac [1] (see also Refs. 2-4,10,11,14,15,26,27) is provided by the substitution of the photon operators, forming a representation of the Weyl-Heisenberg algebra, into the... 

This result can be obtained by canonical quantization of the components of classical angular momentum (9) [2]. Hereafter in this section we use the following notation ... 

Now consider the set of Stokes operators that can be obtained by canonical quantization of (132). On the other hand, the Stokes operators should by definition represent the complete set of independent Hermitian bilinear forms in the photon operators of creation and annihilation. It is clear that such a set is represented by the generators of the SU(3) subalgebra in the Weyl-Heisenberg algebra of electric dipole photons. The nine generators have the form [46]... 

It is also seen that the vacuum fluctuations of the field with m =/= m contribute to the quantum noise of polarization of the mode with given m. Similar results can be obtained for the position dependent Stokes operators obtained from (132) by canonical quantization. 

This Hamiltonian results from the standard canonical quantization of electrodynamics if it is assumed that particle speeds are negligible compared to the speed of light, and all charge-photon interactions are discarded the Coulomb gauge condition must also be imposed [8],... 

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... 

By following the ideas of canonical quantization, both manifest Lorentz co-variance and gauge invariance of the theoretical description have been sacrificed. All physical quantities such as transition amplitudes and S-matrix elements, however, will be Lorentz covariant and independent of the chosen... 

We have considered the stochastic dynamics of a particle interacting with its environment of two-level systems in the presence of an external potential field. The treatment is based on the canonical quantization procedure. This approach directly yields the dissipative term and the noise operators. It may be pertinent to mention that, although the calculation of dissipative effects is straightforward, the treatment of noise is not simple as far as the path integral techniques are concerned. 

Kundt, W. Canonical Quantization of Gau Invariant Field Theories (Vol. 40)... 

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