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Quantum Diffeomorphism Invariance on the Worldline

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Timeless Quantum Mechanics and the Early Universe

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Abstract

We now discuss the quantization of the general theory presented in Chap. 2, and we present a formalism for the construction and interpretation of quantum relational observables by closely following the analogy with the classical theory and, in particular, the HJ formalism.

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Notes

  1. 1.

    This chapter is based on [1,2,3].

  2. 2.

    If \(\Lambda \) is considered as a fixed constant, it acquires the status of law in the sense that it defines the constraint \(C_{\Lambda }\), and its value determines which independent initial conditions on the scalars are allowed by the theory.

  3. 3.

    It is also worthwhile to mention Komar’s approach [9], in which the Dirac conditions (3.4) are imposed, but the constraints are not self-adjoint, and the evolution is defined by an operator which is proportional to the Hermitian conjugate of the constraints.

  4. 4.

    We assume that the labels \({\mathbf {k}}\) are independent of \(p_e\) and E.

  5. 5.

    If the spectrum of \(\hat{C}\) is discrete, it may be possible to obtain a formula that is analogous to (3.20), where the integral over \(\tau \) is performed over a finite interval. For example, if E spans the integers, we can integrate \(\tau \) over the interval \((0,2\pi )\).

  6. 6.

    In [22], on-shell operators were considered with a different notation and they were called ‘projected kernels’ (see Eq. 13.32 in [22]).

  7. 7.

    See also [1, 23].

  8. 8.

    The corresponding classical evolution is dictated by the gauge-fixed total Hamiltonian (2.65) with \(\omega \equiv 1\), such that \(H_T^{{\text {gf}}} = C\). This is referred to as the ‘proper-time gauge’.

  9. 9.

    As was mentioned in Footnote 5, it may be possible to obtain a formula similar to (3.33), where the integral is performed over a finite interval if the spectrum of \(\hat{C}\) is discrete. For instance, if E spans the integers, the integral over \(\tau \) may be performed from 0 to \(2\pi \).

  10. 10.

    The abbreviation “h.c.” is a short-hand for “Hermitian conjugate”. Here, the term Hermitian refers to the auxiliary inner product.

  11. 11.

    Specifically, \(\hat{P}_{t = s-\tau } = \sum _{{\mathbf {k}}}\int {\mathrm {d}}t\ \delta (t-s+\tau )|{t,{\mathbf {k}}}\rangle \langle {t,{\mathbf {k}}}|\).

  12. 12.

    Notice that \(T(\tau ) = s\) implies that \(t = s-\tau \), which is the condition enforced by \(\hat{P}_{t = s-\tau }\).

  13. 13.

    Notice that \(\hat{P}_0[\hat{f},\hat{C}]\hat{P}_{t = t_0}\hat{P}_{E = 0} = \hat{P}_0\hat{f}\hat{C}\hat{P}_{t = t_0}\hat{P}_{E = 0}\).

  14. 14.

    In the particular case in which \(\hat{f} = \hat{1}\), the second line of (3.43) coincides with (3.25).

  15. 15.

    Instead of (3.44), one could consider, in principle, a decomposition in which the eigenvalue f also has an explicit dependence on s. This is not discussed here. See, however, the developments of Sect. 3.5.2.

  16. 16.

    It is important to emphasize that the quantum Faddeev–Popov resolution of the identity given in (3.39) or (3.69) is a key feature of our formalism. In this way, we differ, for example, from the other method proposed by Marolf in [23], where invariant extensions were defined in a way that did not invariantly extend the identity operator to the identity in \(\mathcal {H}_{\text {phys}}\), and thus the Faddeev–Popov resolution of the identity was not reproduced. In the case of the relativistic particle (to be analyzed in Chap. 4), the formalism of [23] leads to \(\hat{\mathcal {O}}_{\text {inv}}[1|q^0 = c s] = \mathrm {sgn}(\hat{p_0}) \ne \hat{1}\). It is our opinion that reproducing the Faddeev–Popov resolution of the identity in the quantum theory should be the correct procedure. It is also worthwhile to note that (3.69) heuristically corresponds to “inserting a gauge condition operator” into the (auxiliary) inner product. This is a procedure that was suggested in [22, 24].

  17. 17.

    Notice that the evolution determined by the classical gauge-fixed Hamiltonian (2.65) is given by \(\{f,H_T^{{\text {gf}}}\} = \{f,\omega C\}\) if f only depends on the scalars.

  18. 18.

    Recall that we have made the simplifying assumption that \(\hat{h}\) does not depend on s.

  19. 19.

    Although it is possible to choose more general (phase-space) functions as generalized clocks, the choice of \(\chi (q)\) will be sufficient in the perturbation theory developed in Chaps. 6 and 7.

  20. 20.

    See, however, [27] for a related discussion.

  21. 21.

    Notice that the physical propagator \((\sigma ,{\mathbf {n}};s|\sigma _0,{\mathbf {n}}_0;s_0)\) is a particular case of the matrix \((\sigma _1,{\mathbf {n}}_1;\chi _1|\sigma _2,{\mathbf {n}}_2;\chi _2)\), but one changes the value of the clock instead of switching clocks.

  22. 22.

    These maps are akin to constructions that had been considered earlier in various contexts, such as those of non-Abelian gauge fields in [33] or quantum canonical transformations in [34].

  23. 23.

    After the release of [2] (on which part of this chapter is based) and its submission by the author of this thesis for publication, a generalization of [19] to “relativistic settings” appeared in [51]. The results therein are complementary to the ones presented in [2] and in the beginning of this section (Sect. 3.7), but, in contrast to [2] and Sect. 3.7, they are still restricted to a particular class of gauge choices (which have trivial Faddeev–Popov determinants). In the future, it would be interesting to compare the formalism presented here to the one proposed in [19, 51].

  24. 24.

    We assume that E and \(E_{(1)}\) are labels of the same type (discrete or continuous) and that the equation \(E = E_{(1)}+E_{(>1)}\) can be solved for \(E_{(1)}\). The solution is given by h. For example, if both E and \(E_{(1)}\) are continuous labels, whereas \(E_{(>1)}\) is discrete, then the restriction of h to the case in which \(E = 0\) corresponds to a restriction of h to a set of discrete values given by \(-E_{(>1)}\).

  25. 25.

    The G-twirl operation is also sometimes used for spatial frames of reference (associated with generalized rods) [31].

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Authors and Affiliations

  1. Institut für Theoretische Physik, Universität zu Köln, Cologne, Germany

    Leonardo Chataignier

  2. Dipartimento di Fisica e Astronomia, Università di Bologna, Bologna, Italy

    Leonardo Chataignier

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Chataignier, L. (2022). Quantum Diffeomorphism Invariance on the Worldline. In: Timeless Quantum Mechanics and the Early Universe. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-94448-3_3

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