Abstract
This will be a very hard chapter. Not because of abstract and technically advanced mathematics, which can easily be learned as soon as the underlying physics is clear, that is, as soon as the need for abstraction is evident. It will be hard for two reasons. First, there does not exist a fundamental, mathematically coherent and consistent formulation of a relativistic quantum theory with interaction that could extend the analysis of the foregoing chapters to relativistic physics. We shall say more about that in a moment.
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Notes
- 1.
- 2.
See T. Maudlin, Quantum Non-Locality and Relativity, 3rd edn, Wiley-Blackwell, 2011.
- 3.
We do not wish to further scrutinize the assumption of a point particle. It is sufficient here to understand that a point has a relativistically invariant form. There are other relativistically invariant forms, but so far considerations along such lines have not led to empirically adequate formulations.
- 4.
If the field A is not zero, then there is in general no longer a spectral gap in (−mc 2, mc 2).
- 5.
One may turn the argument around and take the avoidance of the radiation catastrophe as justification for the Pauli exclusion principle. In fact, along these lines, a generalization of the Pauli exclusion principle was later established, known as the spin–statistics theorem, which states that many-particle half integer spin wave functions must be antisymmetric, i.e., fermionic wave functions (see Remark 4.9).
- 6.
See, e.g., D.-A. Deckert, D. Dürr, F. Merkl, M. Schottenloher, Time-evolution of the external field problem in quantum electrodynamics. J. Math. Phys. 51 (12), 122301 (2010); arXiv:0906.0046, for more details.
- 7.
For a mathematically precise formulation, see D.-A. Deckert, D. Dürr, F. Merkl, M. Schottenloher, Time-evolution of the external field problem in quantum electrodynamics. J. Math. Phys. 51 (12), 122301 (2010), arXiv:0906.0046; D. Lazarovici, Time evolution in the external field problem of quantum electrodynamics. Thesis, LMU Munich, 2011, online version arXiv:1310.1778; D.-A. Deckert and F. Merkl, External field QED on Cauchy surfaces for varying electromagnetic fields, Commun. Math. Phys. 345 (3), 973–1017 (2016); arXiv:1505.06039.
- 8.
A general result like this is known in quantum field theory as Haag’s theorem. It can be formulated, for example, by stating that free and interacting field operators lead to representations of the canonical commutation relations which are not unitarily equivalent. In the Dirac sea picture, this abstract result can be vividly and physically interpreted in terms of the creation of infinitely many pairs.
- 9.
This is a natural generalisation of the directional derivative of functions of many variables, in which the derivative is a linear map of directional vectors to real values (viewed geometrically as the value of a slope on a surface). In short, 〈∇F(x), h〉 is the derivative of F in the direction h at the point x. In the present case, we have an uncountable set of variables and the directional vector becomes a function, while the scalar product sum is replaced by an integral. The definition is thus, for “test functions” h,
$$\displaystyle \begin{aligned} \int \frac{\updelta H(f)}{\updelta f}\bigg\lvert_{f=F(\mathbf x)} h(\mathbf x)\,\mathrm{d}^n x= \lim_{\varepsilon\to 0} \frac{H\big(F(\mathbf x) +\varepsilon h(\mathbf x)\big)-H\big(F(\mathbf x)\big)}{\varepsilon}\,. \end{aligned}$$ - 10.
In fact, the trajectories are Hölder-continuous with exponent 1/2, i.e., |B(t) − B(s)|∼|t − s|1∕2.
- 11.
D.-A. Deckert, M. Esfeld, and A. Oldofredi, A persistent particle ontology for QFT in terms of the Dirac sea. In the British Journal for the Philosophy of Science, online version: arXiv:1608.06141, 2016.
- 12.
D. Lazarovici, Time evolution in the external field problem of quantum electrodynamics. Thesis, LMU Mnchen, 2011. Online version: arXiv:1310.1778.
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Dürr, D., Lazarovici, D. (2020). Relativistic Quantum Theory. In: Understanding Quantum Mechanics . Springer, Cham. https://doi.org/10.1007/978-3-030-40068-2_11
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DOI: https://doi.org/10.1007/978-3-030-40068-2_11
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