Abstract
This note describes the restoration of time in one-dimensional parameterization-invariant (hence timeless) models, namely, the classically equivalent Jacobi action and gravity coupled to matter. It also serves as a timely introduction by examples to the classical and quantum BV-BFV formalism as well as to the AKSZ method.
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Notes
To be more precise, the 1 form \(\check{\alpha }\) is in general defined on the space \(\check{F}^\partial \) of jets of the fields at a boundary point. The space of boundary fields \(F^\partial \) is then defined as the reduction of \(\check{F}^\partial \) by the kernel of \(\check{\omega }\) and is assumed to be smooth. The relation \(L_{[a,b]}\) is then automatically isotropic (i.e., the restriction of \(\omega \) to it is zero) and is assumed to be Lagrangian (i.e., of maximal dimension) for the theory to be well defined. The last condition in particular implies that C is coisotropic (i.e., that it is locally defined by constraints whose Poisson bracket vanishes on C, a.k.a. first class constraints). In this paper all the \(F^\partial \)’s are smooth manifolds and all the \(L_{[a,b]}\)’s are Lagrangian submanifolds.
Note that \(L_{[a,b]}\) is not a graph since on the source \(T^*U\) one has to impose the constraint \(\frac{||p||^2}{2m}+V(q) = E\). On the other hand, since the flow is for a nonfixed time s, we have that anyway \(\dim L_{[a,b]}=2n\) and \(L_{[a,b]}\) is indeed Lagrangian.
The attentive reader might have noticed that the Jacobi theory actually has a smaller phase space as one has to remove values corresponding to singularities of the Lagrangian function (velocity equals zero in the example \(V=0\)). This shows that one-dimensional gravity is an improved version of the Jacobi theory that cures the singularities.
The BV action in the present example is linear in the antifields, which is not a general feature of the formalism.
As \(\xi ^+\) is even, we could, however, impose \(\xi ^+\not =0\) to make the fourth equation nonsingular and solvable for \(\Xi \). The fifth equation would then be automatically satisfied.
To be precise, the result is proved for finite-dimensional \(\mathcal {F}\). Note that being a pullback, the BV form on \(\mathcal {F}\) is necessarily degenerate, unless \(\mathcal {B}\) is zero-dimensional; it is assumed anyway that the BV form on \(\mathcal {Y}\) is nondenegerate. In the infinite-dimensional case, however, it is possible (and it turns out to be true in examples) that both BV forms, on \(\mathcal {F}\) and \(\mathcal {Y}\), are weakly nondegenerate. The general proof of the modified quantum master equation, however, becomes formal (note that, to start with, \(\Delta \) is not even well defined). The correct viewpoint is that one expects the modified quantum master equation to hold, but one has to prove it explicitly from the perturbation theory.
This is the case when the space of backgrounds is affine. In more general situations, the splitting actually requires a choice of coordinates around each point and one has to resort to formal geometry, see [7] and references therein.
We only allow nontrivial backgrounds for the even fields. In other words, we set the background of all odd fields to zero. The minimal space of residual fields is the tangent space, also in the odd directions, of the space of background fields at such a point.
We may also choose the polarization in which states are functions of q and b. Then, c is quantized as \(-{\mathrm {i}}\hbar \frac{\partial }{\partial b}\). A state \(\phi \) will then be of the form \(\phi _{-1}b+\phi _0\) \(\phi _{-1},\phi _0\in L^2(\mathbb {R}^n)\). Cohomology in degree zero is now the cokernel of \(\widehat{H}_E\), which in a Hilbert space can be canonically identified with the kernel of \(\widehat{H}_E\).
From now, we use Einstein’s summation convention.
Note that the only thing that matters is that G is nondegenerate, but otherwise, it can have any signature. We present examples both with Euclidean and with Lorentzian metrics.
Equivalently, one may change variables \(p\mapsto p+A\) and remove A from f at the price of getting the one-form \((p+A)\cdot \mathrm {d}q+b\,\mathrm {d}c\). This is a better formulation if A is a connection but not a globally well-defined form.
References
Alexandrov, M., Kontsevich, M., Schwarz, A., Zaboronsky, O.: “The geometry of the master equation and topological quantum field theory. Int. J. Mod. Phys. A12, 1405–1430 (1997)
Barbour, J.: The nature of time. arXiv:0903.3489
Batalin, I.A., Fradkin, E.S.: A generalized canonical formalism and quantization of reducible gauge theories. Phys. Lett. B 122, 157–164 (1983)
Batalin, I.A., Vilkovisky, G.A.: Gauge algebra and quantization. Phys. Lett. B 102, 27–31 (1981)
Batalin, I.A., Vilkovisky, G.A.: Relativistic S-matrix of dynamical systems with boson and fermion costraints. Phys. Lett. B 69, 309–312 (1977)
Becchi, C., Rouet, A., Stora, R.: Renormalization of the abelian Higgs-Kibble model. Commun. Math. Phys. 42, 127–162 (1975)
Bonechi, F., Cattaneo, A.S., Mnëv, P.: The Poisson sigma model on closed surfaces. JHEP 2012(99), 1–27 (2012)
Brink, L., Deser, S., Zumino, B., Di Vecchia, P., Howe, P.: Local supersymmetry for spinning particles. Phys. Lett. B 64, 435–438 (1976)
Cattaneo, A.S., Mnëv, P., Reshetikhin, N.: Classical BV theories on manifolds with boundaries. Commun. Math. Phys. 332, 535–603 (2014)
Cattaneo, A.S., Mnëv, P., Reshetikhin, N.: Classical and quantum Lagrangian field theories with boundary. In: Proceedings of the Corfu Summer Institute 2011 School and Workshops on Elementary Particle Physics and Gravity. arXiv:1207.0239
Cattaneo, A.S.: Mnëv, P., Reshetikhin, N.: Perturbative quantum gauge theories on manifolds with boundary. Commun. Math. Phys. arXiv:1507.01221
Cattaneo, A.S., Schiavina, M.: BV-BFV approach to general relativity: Einstein–Hilbert action. J. Math. Phys. 57, 023515, 17 (2016)
Cattaneo, A.S., Schiavina, M.: BV-BFV approach to general relativity II: Palatini–Holst action (in preparation)
Geztler, E.: The Batalin-Vilkovisky formalism of the spinning particle. arXiv:1511.02135
Geztler, E.: The spinning particle with curved target. arXiv:1605.04762
Hartle, J.B., Hawking, S.W.: Wave function of the Universe. Phys. Rev. D 28, 2960 (1983)
Jacobi, C.G.J.: Das Princip der kleinsten Wirkung. Ch. 6. In: Clebsch, A. (ed.) Vorlesungen über Dynamik. Available at Gallica-Math. Reimer, Berlin (1866)
Kijowski, J., Tulczyjew, M.: A Symplectic Framework for Field Teories, Lecture Notes in Physics, vol. 107. Springer, New york (1979)
Kostant, B., Sternberg, S.: Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras. Ann. Phys. 176, 49–113 (1987)
Roytenberg, D.: AKSZ-BV formalism and courant algebroid-induced topological field theories. Lett. Math. Phys. 79, 143–159 (2007)
Schiavina, M.: BV-BFV Approach to general relativity, PhD thesis, Zurich (2015)
Tyutin, I.V.: Gauge invariance in field theory and statistical physics in operator formalism. Lebedev Institute preprint No. 39 (1975). arXiv:0812.0580
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We thank the referee for a number of very valuable comments.
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We acknowledge partial support of SNF Grant No. 200020-149150/1. This research was (partly) supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation, the COST Action MP1405 QSPACE, and COST (European Cooperation in Science and Technology).
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Cattaneo, A.S., Schiavina, M. On Time. Lett Math Phys 107, 375–408 (2017). https://doi.org/10.1007/s11005-016-0907-x
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DOI: https://doi.org/10.1007/s11005-016-0907-x
Keywords
- Parametrization invariant Lagrangian
- Jacobi action
- One-dimensional gravity
- BV
- BFV
- AKSZ
- Spinning particle
- Supersymmetry