# Precanonical Structure of the Schrödinger Wave Functional of a Quantum Scalar Field in Curved Space-Time

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Quantum Scalar Field on a Curved Space-Time: The Canonical and Precanonical Descriptions

**Ψ**, ${x}^{\mu}=(t,\mathbf{x})=(t,{x}^{i})$ are space-time coordinates, ${g}_{\mu \nu}$ is the space-time metric tensor whose components depend on ${x}^{\mu}$, $g=|det({g}_{\mu \nu}\left)\right|$. In this equation, one uses the space-time coordinates adapted to the space-like foliation such as the induced metric on the space-like leaves of the foliation is ${g}_{ij}$, the lapse $N=\sqrt{{g}_{00}}$ and the shift functions ${N}_{i}={g}_{0i}=0$.

## 3. Relating the Precanonical Wave Function and the Schrödinger Wave Functional

#### 3.1. The Restriction of Precanonical Schrödinger Equation to $\Sigma $

#### 3.1.1. The Total Covariant Derivative

#### 3.2. The Time Evolution of the Schrödinger Wave Functional from pSE

#### 3.3. The Functional Derivatives of $\mathbf{\Psi}$

#### 3.4. The Correspondence between Terms I–V in Equation (14) and the Canonical Hamiltonian Operator in (1)

#### 3.4.1. The Potential Term V

#### 3.4.2. The Second Functional Derivative Term

#### 3.4.3. The Non-Ultralocality Term and the Wave Functional $\mathbf{\Psi}$ in Terms of Precanonical ${\Psi}_{\Sigma}$

#### 3.4.4. The Vanishing Contribution from the Terms I and $IIIa$

- (i)
- the first boundary term is the result of the covariant Stokes theorem and it vanishes if ${\Psi}_{\Sigma}$ vanishes on the boundary $\partial \Sigma $, i.e., the spatial infinity;
- (ii)
- the following three terms follow from the Leibniz rule for the total covariant derivative ${\nabla}_{i}^{tot}$ with respect to the Clifford products of tensor Clifford-algebra-valued functions;
- (iii)
- in the second term, ${\nabla}_{i}^{tot}\left({\gamma}_{0}{\gamma}^{i}\right)=0$ due to the covariant constancy of Dirac matrices (12);
- (iv)
- in the third term, the metric compatibility yields ${\nabla}_{i}\sqrt{h}=0$;
- (v)
- in the fourth term, the explicit formula for $\mathbf{\Phi}\left(\mathbf{x}\right)$ in (31) yields$${\nabla}_{i}^{tot}\mathbf{\Phi}\left(\mathbf{x}\right)=\frac{-\mathrm{i}}{\varkappa}\mathbf{\Phi}\left(\mathbf{x}\right)\left(\right)open="("\; close=")">{\partial}_{i}\varphi {\gamma}^{l}{\partial}_{l}\varphi +\varphi {\gamma}^{l}{\partial}_{il}\varphi +\varphi \left({\nabla}_{i}^{tot}{\gamma}^{l}\right){\partial}_{l}\varphi $$$$\begin{array}{c}\hfill \begin{array}{c}\hfill \phantom{\rule{-10.0pt}{0ex}}\int \phantom{\rule{-0.166667em}{0ex}}\mathrm{d}\mathbf{x}\phantom{\rule{4pt}{0ex}}\mathrm{Tr}\left(\right)open="\{"\; close="\}">\mathbf{\Phi}\left(\mathbf{x}\right)\frac{1}{\varkappa}{\gamma}_{0}{\Psi}_{\Sigma}\left(\mathbf{x}\right)\left(\right)open="("\; close=")">{g}^{il}{\partial}_{i}\varphi {\partial}_{l}\varphi +\varphi {g}^{il}{\partial}_{il}\varphi \end{array}=\mathbf{\Psi}\int \phantom{\rule{-0.166667em}{0ex}}\mathrm{d}\mathbf{x}\phantom{\rule{4pt}{0ex}}\sqrt{g}\left(\right)open="("\; close=")">{g}^{il}{\partial}_{i}\varphi {\partial}_{l}\varphi +\varphi {g}^{il}{\partial}_{il}\varphi \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =-\mathbf{\Psi}\int \phantom{\rule{-0.166667em}{0ex}}\mathrm{d}\mathbf{x}\phantom{\rule{4pt}{0ex}}\sqrt{h}{\nabla}_{i}^{}\left(\right)open="("\; close=")">\sqrt{{g}_{00}}{g}^{il}\frac{1}{2}{\partial}_{l}{\varphi}^{2}=0,\hfill \end{array}$$

## 4. Static Space-Times with ${\mathbf{\omega}}_{\mathbf{0}}=\mathbf{0}$

`prodint`) is used

## 5. Non-Static Space-Times with ${\mathbf{\omega}}_{\mathbf{0}}\ne \mathbf{0}$

## 6. Conclusions

- (i)
- (ii)
- (iii)
- The required cancellation of one of those additional terms with a similar term $II$ in (14), which also has no obvious counterpart in (1), leads to the expression of the Schrödinger wave functional as the trace of the continuous product of restricted precanonical wave functions (40) over all spatial points, which we later interpret as a multidimensional product integral, Equation (41);
- (iv)
- The expression of the wave functional in terms of precanonical wave functions substituted into the second additional term mentioned in (ii) reproduces the second term in the right-hand side of (1), which is responsible for non-ultralocality;
- (v)
- The expression of the wave functional in terms of precanonical wave functions obtained in (iii) also implies that under the boundary conditions of vanishing fields $\varphi \left(\mathbf{x}\right)$ and ${\Psi}_{\Sigma}(\varphi \left(\mathbf{x}\right),\mathbf{x},t)$ at the spatial infinity the terms I and $IIIa$ in (14) do not contribute to the canonical Schrödinger equation (1);
- (vi)
- As a consequence of (i)–(v), in static space-times with ${\omega}_{0}=0$, the functional Schrödinger equation (1) is thus derived from the precanonical Schrödinger equation (2) and the Schrödinger wave functional is expressed as the trace of the product integral of precanonical wave functions;
- (vii)
- In non-static space-times with ${\omega}_{0}\ne 0$, the transformation (55) absorbs the contribution of the term $IIIb$ in (14) thus allowing us again to obtain the functional Schrödinger equation (1) from the precanonical Schrödinger equation (2) and to express the Schrödinger wave functional in terms of transformed precanonical wave functions;
- (viii)

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Birrell, N.; Davies, P. Quantum Fields in Curved Space; Cambridge University Press: Cambridge, UK, 1982. [Google Scholar]
- Wald, R.M. Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics; University of Chicago Press: Chicago, IL, USA, 1994. [Google Scholar]
- Fulling, S.A. Aspects of Quantum Field Theory in Curved Space-Time; Cambridge University Press: Cambridge, UK, 1989. [Google Scholar]
- Parker, L.; Toms, D. Quantum Field Theory in Curved Spacetime; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Benini, M.; Dappiaggi, C.; Hack, T.-P. Quantum field theory on curved backgrounds—A primer. Int. J. Mod. Phys. A
**2013**, 28, 1330023. [Google Scholar] [CrossRef] - Kanatchikov, I.V. Towards the Born-Weyl quantization of fields. Int. J. Theor. Phys.
**1998**, 37, 333–342. [Google Scholar] [CrossRef] - Kanatchikov, I.V. De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory. Rep. Math. Phys.
**1999**, 43, 157–170. [Google Scholar] [CrossRef] - Kanatchikov, I.V. On quantization of field theories in polymomentum variables. AIP Conf. Proc.
**1998**, 453, 356–367. [Google Scholar] - Kanatchikov, I.V. Geometric (pre)quantization in the polysymplectic approach to field theory. In Differential Geometry and Its Applications; Krupka, D., Kowalski, O., Krupkova, O., Slovák, J., Eds.; Worlds Scientific: Singapore, 2008; pp. 309–322. [Google Scholar]
- De Donder, T. Théorie Invariantive du Calcul des Variations; Gauthier-Villars: Paris, France, 1935. [Google Scholar]
- Weyl, H. Geodesic fields in the calculus of variations. Ann. Math.
**1935**, 36, 607–629. [Google Scholar] [CrossRef] - Rund, H. The Hamilton-Jacobi Theory in the Calculus of Variations; D. Van Nostrand: Toronto, ON, Canada, 1966. [Google Scholar]
- Kastrup, H. Canonical theories of Lagrangian dynamical systems in physics. Phys. Rep.
**1983**, 101, 1–167. [Google Scholar] [CrossRef] - Gotay, M.J.; Isenberg, J.; Marsden, J.E. Momentum maps and classical relativistic fields I: Covariant field theory. arXiv
**2004**, arXiv:physics/9801019. [Google Scholar] - Román-Roy, N. Some properties of multisymplectic manifolds. In Classical and Quantum Physics; Marmo, G., Martin de Diego, D., Muñoz-Lecanda, M., Eds.; Springer Nature: Basel, Switzerland, 2019; pp. 325–336. [Google Scholar]
- Forger, M.; Gomes, L.G. Multisymplectic and polysymplectic structures on fiber bundles. Rev. Math. Phys.
**2013**, 25, 1350018. [Google Scholar] [CrossRef] - Forger, M.; Yepes, Z. Lagrangian distributions and connections in multisymplectic and polysymplectic geometry. Differ. Geom. Appl.
**2013**, 31, 775–807. [Google Scholar] [CrossRef] - Román-Roy, N. Multisymplectic Lagrangian and Hamiltonian formalisms of classical field theories. Symmetry Integr. Geom. Methods Appl.
**2009**, 5, 100. [Google Scholar] [CrossRef] - Bridges, T.J.; Hydon, P.E.; Lawson, J.K. Multisymplectic structures and the variational bicomplex. Math. Proc. Camb. Philos. Soc.
**2010**, 148, 159–178. [Google Scholar] [CrossRef] - McLean, M.; Norris, L.K. Covariant field theory on frame bundles of fibered manifolds. J. Math. Phys.
**2000**, 41, 6808. [Google Scholar] [CrossRef] - Norris, L.K. n-symplectic algebra of observables in covariant Lagrangian field theory. J. Math. Phys.
**2001**, 42, 4827. [Google Scholar] [CrossRef] - Günther, C. The polysymplectic Hamiltonian formalism in field theory and calculus of variations: I. The local case. J. Differ. Geom.
**1987**, 25, 23–53. [Google Scholar] [CrossRef] - Sardanashvily, G. Polysymplectic Hamiltonian field theory. arXiv
**2015**, arXiv:1505.01444. [Google Scholar] - McClain, T. Some Considerations in the Quantization of General Relativity. Ph.D. Thesis, The University of Texas at Austin, Austin, TX, USA, 2018. [Google Scholar] [CrossRef]
- Blacker, C. Quantization of polysymplectic manifolds. J. Geom. Phys.
**2019**, 145, 103480. [Google Scholar] [CrossRef] - Awane, A.; Goze, M. Pfaffian Systems, k-Symplectic Systems; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2000. [Google Scholar]
- Struckmeier, J.; Redelbach, A. Covariant Hamiltonian field theory. Int. J. Mod. Phys. E
**2008**, 17, 435–491. [Google Scholar] [CrossRef] - Krupkova, O. Hamiltonian field theory. J. Geom. Phys.
**2002**, 43, 93. [Google Scholar] [CrossRef] - Zatloukal, V. Classical field theories from Hamiltonian constraint: Local symmetries and static gauge fields. Adv. Appl. Clifford Algebras
**2018**, 28, 48. [Google Scholar] [CrossRef] - Fernandes, M.C.B. DKP covariant Hamiltonian dynamics for antisymmetric tensor fields. Rep. Math. Phys.
**2020**, 85. to appear. [Google Scholar] - de León, M.; Salgado, M.; Vilariño, S. Methods of Differential Geometry in Classical Field Theories: k-Symplectic and k-Cosymplectic Approaches; World Scientific: Singapore, 2005. [Google Scholar]
- Kanatchikov, I.V. On the canonical structure of the De Donder-Weyl covariant Hamiltonian formulation of field theory I. Graded Poisson brackets and equations of motion. arXiv
**1993**, arXiv:hep-th/9312162. [Google Scholar] - Kanatchikov, I.V. From the Poincaré-Cartan Form to a Gerstenhaber Algebra of Poisson Brackets in Field Theory. In Quantization, Coherent States, and Complex Structures; Antoine, J.P., Ali, S.T., Lisiecki, W., Mladenov, I.M., Odzijewicz, A., Eds.; Springer: Boston, MA, USA, 1995; pp. 173–183. [Google Scholar]
- Kanatchikov, I.V. Canonical structure of classical field theory in the polymomentum phase space. Rep. Math. Phys.
**1998**, 41, 49–90. [Google Scholar] [CrossRef] - Kanatchikov, I.V. On field theoretic generalizations of a Poisson algebra. Rep. Math. Phys.
**1997**, 40, 225–234. [Google Scholar] [CrossRef] - Forger, M.; Romero, S. Covariant Poisson brackets in geometric field theory. Commun. Math. Phys.
**2005**, 256, 375–410. [Google Scholar] [CrossRef] - Forger, M.; Paufler, C.; Römer, H. The Poisson bracket for Poisson forms in multisymplectic field theory. Rev. Math. Phys.
**2003**, 15, 705–744. [Google Scholar] [CrossRef] - Kanatchikov, I.V. On the Duffin-Kemmer-Petiau formulation of the covariant Hamiltonian dynamics in field theory. Rep. Math. Phys.
**2000**, 46, 107–112. [Google Scholar] [CrossRef] - Cantrijn, F.; Ibort, A.; de León, M. On the geometry of multisymplectic manifolds. J. Austral. Math. Soc. Ser. A
**1999**, 66, 303–330. [Google Scholar] [CrossRef] - Rogers, C.L. L
_{∞}-algebras from multisymplectic geometry. Lett. Math. Phys.**2012**, 100, 29–50. [Google Scholar] [CrossRef] - Baez, J.C.; Rogers, C.L. Categorified symplectic geometry and the string Lie 2-algebra. Homol. Homotopy Appl.
**2010**, 12, 221–236. [Google Scholar] [CrossRef] - Fiorenza, D.; Roger, C.L.; Schreiber, U. L
_{∞}-algebras of local observables from higher prequantum bundles. Homol. Homotopy Appl.**2014**, 16, 107–142. [Google Scholar] [CrossRef] - Richter, M. Towards homotopy Poisson-n algebras from N-plectic structures. arXiv
**2018**, arXiv:1506.01129. [Google Scholar] - Ryvkin, L.; Wurzbacher, T. An invitation to multisymplectic geometry. J. Geom. Phys.
**2019**, 142, 9–36. [Google Scholar] [CrossRef] - Kanatchikov, I.V. Covariant geometric prequantization of fields. arXiv
**2001**, arXiv:gr-qc/0012038. [Google Scholar] - Hélein, F.; Kouneiher, J. The notion of observable in the covariant Hamiltonian formalism for the calculus of variations with several variables. Adv. Theor. Math. Phys.
**2004**, 8, 735–777. [Google Scholar] [CrossRef] - Hélein, F.; Kouneiher, J. Covariant Hamiltonian formalism for the calculus of variations with several variables. Adv. Theor. Math. Phys.
**2004**, 8, 565–601. [Google Scholar] [CrossRef] - Kanatchikov, I.V. Novel algebraic structures from the polysymplectic form in field theory. arXiv
**1997**, arXiv:hep-th/9612255. [Google Scholar] - Kanatchikov, I. On a generalization of the Dirac bracket in the De Donder-Weyl Hamiltonian formalism. In Differential Geometry and Its Applications; Kowalski, O., Krupka, D., Krupková, O., Slovák, J., Eds.; World Scientific: Singapore, 2008; pp. 615–625. [Google Scholar]
- Lachiéze-Rey, M. Historical Hamiltonian dynamics: Symplectic and covariant. arXiv
**2016**, arXiv:1602.07281. [Google Scholar] - Kaminaga, Y. Poisson bracket and symplectic structure of covariant canonical formalism of fields. Electron. J. Theor. Phys.
**2018**, 14, 55–72. [Google Scholar] - Castellani, L.; D’Adda, A. Covariant hamiltonian for gravity coupled to p-forms. arXiv
**2019**, arXiv:1906.11852. [Google Scholar] - Vey, D. Multisymplectic formulation of vielbein gravity: I. De Donder-Weyl formulation, Hamiltonian (n − 1)-forms. Class. Quant. Grav.
**2015**, 32, 095005. [Google Scholar] [CrossRef] - Hrabak, S.P. On a multisymplectic formulation of the classical BRST symmetry for first order field theories: I. Algebraic structures. arXiv
**1999**, arXiv:math-ph/9901012. [Google Scholar] - Berra-Montiel, J.; del Río, E.; Molgado, A. Polysymplectic formulation for topologically massive Yang–Mills field theory. Int. J. Mod. Phys. A
**2019**, 32, 1750101. [Google Scholar] [CrossRef][Green Version] - Berra-Montiel, J.; Molgado, A.; Serrano-Blanco, D. De Donder–Weyl Hamiltonian formalism of MacDowell–Mansouri gravity. Class. Quant. Grav.
**2017**, 34, 235002. [Google Scholar] [CrossRef][Green Version] - Berra-Montiel, J.; Molgado, A.; Rodríguez–López, A. Polysymplectic formulation for BF gravity with Immirzi parameter. Class. Quant. Grav.
**2019**, 36, 115003. [Google Scholar] [CrossRef][Green Version] - de León, M.; Prieto-Martínez, P.D.; Román-Roy, N.; Vilariño, S. Hamilton-Jacobi theory in multisymplectic classical field theories. J. Math. Phys.
**2017**, 58, 092901. [Google Scholar] [CrossRef][Green Version] - de León, M.; Vilariño, S. Hamilton-Jacobi theory in k-cosymplectic field theories. Int. J. Geom. Meth. Mod. Phys.
**2014**, 11, 1450007. [Google Scholar] [CrossRef][Green Version] - de León, M.; Marrero, J.C.; de Diego, D.M. A geometric Hamilton-Jacobi theory for classical field theories. In Variations, Geometry and Physics; Krupkova, O., Saunders, D., Eds.; Nova Science Publishers, Inc.: New York, NY, USA, 2009; pp. 129–140. [Google Scholar]
- Fulp, R.O.; Lawson, J.K.; Norris, L.K. Geometric prequantization on the spin bundle based on n-symplectic geometry: The Dirac equation. Int. J. Theor. Phys.
**1994**, 33, 1011–1028. [Google Scholar] [CrossRef] - Navarro, M. Toward a finite-dimensional formulation of quantum field theory. Found. Phys. Lett.
**1998**, 11, 585. [Google Scholar] [CrossRef][Green Version] - von Hippel, G.M.; Wohlfarth, M. Covariant canonical quantization. Eur. Phys. J. C
**2006**, 47, 861–872. [Google Scholar] [CrossRef] - Kisil, V.V. p-mechanics and De Donder–Weyl theory. Proc. Inst. Math. NAS Ukraine
**2004**, 50, 1108–1115. [Google Scholar] - Nikolic, H. Strings, world-sheet covariant quantization and Bohmian mechanics. Eur. Phys. J. C
**2006**, 47, 525–529. [Google Scholar] [CrossRef][Green Version] - Bashkirov, D.; Sardanashvily, G. Covariant Hamiltonian field theory: Path integral quantization. Int. J. Theor. Phys.
**2004**, 43, 1317–1333. [Google Scholar] [CrossRef][Green Version] - Sardanashvily, G. Deformation quantization in covariant Hamiltonian field theory. arXiv
**2002**, arXiv:hep-th/0203044. [Google Scholar] - Berra-Montiel, J.; Molgado, A.; Serrano-Blanco, D. Moyal product for (n − 1)-forms within the covariant Hamiltonian formalism for fields. J. Phys. Conf. Ser.
**2018**, 1030, 012002. [Google Scholar] [CrossRef] - Cremaschini, C.; Tessarotto, M. Hamiltonian approach to GR—Part 1: Covariant theory of classical gravity. Eur. Phys. J. C
**2017**, 77, 329. [Google Scholar] [CrossRef][Green Version] - Cremaschini, C.; Tessarotto, M. Hamiltonian approach to GR—Part 2: Covariant theory of quantum gravity. Eur. Phys. J. C
**2017**, 77, 330. [Google Scholar] [CrossRef][Green Version] - Kanatchikov, I.V. Ehrenfest theorem in precanonical quantization of fields and gravity. In The Fourteenth Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories, Part C; Bianchi, M., Jantzen, R.T., Ruffini, R., Eds.; World Scientific: Singapore, 2018; pp. 2828–2835. [Google Scholar]
- Kanatchikov, I.V. Precanonical quantum gravity: Quantization without the space-time decomposition. Int. J. Theor. Phys.
**2001**, 40, 1121–1249. [Google Scholar] [CrossRef][Green Version] - Kanatchikov, I.V. From the De Donder-Weyl Hamiltonian Formalism to Quantization of Gravity. arXiv
**1998**, arXiv:gr-qc/9810076. [Google Scholar] - Kanatchikov, I.V. Quantization of Gravity: Yet Another Way. arXiv
**1999**, arXiv:gr-qc/9912094. [Google Scholar] - Kanatchikov, I.V. Precanonical Perspective in Quantum Gravity. Nucl. Phys. Proc. Suppl.
**2000**, 88, 326–330. [Google Scholar] [CrossRef][Green Version] - Kanatchikov, I.V. On the “spin-connection foam” picture of quantum gravity from precanonical quantization. arXiv
**2015**, arXiv:1512.09137. [Google Scholar] - Kanatchikov, I.V. On precanonical quantization of gravity in spin connection variables. AIP Conf. Proc.
**2012**, 1514, 73–76. [Google Scholar] - Kanatchikov, I.V. De Donder-Weyl Hamiltonian formulation and precanonical quantization of vielbein gravity. J. Phys. Conf. Ser.
**2013**, 442, 012041. [Google Scholar] [CrossRef][Green Version] - Kanatchikov, I.V. On precanonical quantization of gravity. Nonlinear Phenom. Complex Syst. (NPCS)
**2014**, 17, 372–376. [Google Scholar] - Kiefer, C. Quantum Gravity; Oxford University Press: Oxford, UK, 2012. [Google Scholar]
- Riahi, N. On the relation between the canonical Hamilton-Jacobi equation and the De Donder-Weyl Hamilton-Jacobi formulation in general relativity. Acta Phys. Pol. B
**2020**, 51. to appear. [Google Scholar] - Hořava, P. On a covariant Hamilton-Jacobi framework for the Einstein-Maxwell theory. Class. Quant. Grav.
**1991**, 8, 2069–2084. [Google Scholar] [CrossRef] - Peres, A. On Cauchy’s problem in general relativity—II. Nuovo Cim.
**1962**, 26, 53–62. [Google Scholar] [CrossRef] - Rovelli, C. Quantum Gravity; Cambridge Univeraity Press: Cambridge, UK, 2004. [Google Scholar]
- Thiemann, T. Modern Canonical Quantum General Relativity; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Kanatchikov, I.V. Ehrenfest theorem in precanonical quantization. J. Geom. Symmetry Phys.
**2015**, 37, 43–66. [Google Scholar] - Finkelstein, D.; Jauch, J.M.; Schiminovich, S.; Speiser, D. Foundations of quaternion quantum mechanics. J. Math. Phys.
**1962**, 3, 207–220. [Google Scholar] [CrossRef] - Finkelstein, D.R.; Galiautdinov, A. Clifford algebra as quantum language. J. Math. Phys.
**2001**, 42, 1489. [Google Scholar] - Adler, S.L. Quaternionic Quantum Mechanics and Quantum Fields; Oxford University Press: New York, NY, USA, 1995. [Google Scholar]
- Horwitz, L. Hypercomplex quantum mechanics. Found. Phys.
**1996**, 26, 851–862. [Google Scholar] [CrossRef][Green Version] - Khrennikov, A. Contextual Approach to Quantum Formalism; Springer: New York, NY, USA, 2009. [Google Scholar]
- Hassanabadi, H.; Sobhani, H.; Banerjee, A. Relativistic scattering of fermions in quaternionic quantum mechanics. Eur. Phys. J. C
**2017**, 77, 581. [Google Scholar] [CrossRef][Green Version] - Moretti, V.; Oppio, M. Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one. Rev. Math. Phys.
**2018**, 31, 1950013. [Google Scholar] [CrossRef][Green Version] - Bolokhov, P.A. Quaternionic wave function. Int. J. Mod. Phys. A
**2019**, 34, 1950001. [Google Scholar] [CrossRef] - Procopio, L.M.; Rozema, L.A.; Wong, Z.J.; Hamel, D.R.; O’Brien, K.; Zhang, X.; Dakić, B.; Walther, P. Single-photon test of hyper-complex quantum theories using a metamaterial. Nat. Comm.
**2017**, 8, 15044. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kanatchikov, I.V. Precanonical quantization of Yang-Mills fields and the functional Schrödinger representation. Rep. Math. Phys.
**2004**, 53, 181–193. [Google Scholar] [CrossRef][Green Version] - Kanatchikov, I.V. On the spectrum of DW Hamiltonian of quantum SU(2) gauge field. Int. J. Geom. Meth. Mod. Phys.
**2017**, 14, 1750123. [Google Scholar] [CrossRef][Green Version] - Kanatchikov, I.V. Schrödinger wave functional in quantum Yang-Mills theory from precanonical quantization. Rep. Math. Phys.
**2018**, 82, 373. [Google Scholar] [CrossRef][Green Version] - Kanatchikov, I.V. Precanonical quantization and the Schrödinger wave functional. Phys. Lett. A
**2001**, 283, 25–36. [Google Scholar] [CrossRef][Green Version] - Kanatchikov, I.V. Precanonical quantization and the Schrödinger wave functional revisited. Adv. Theor. Math. Phys.
**2014**, 18, 1249–1265. [Google Scholar] [CrossRef][Green Version] - Kanatchikov, I.V. On the precanonical structure of the Schrödinger wave functional. Adv. Theor. Math. Phys.
**2016**, 20, 1377–1396. [Google Scholar] [CrossRef][Green Version] - Kanatchikov, I.V. Schrödinger wave functional of a quantum scalar field in static space-times from precanonical quantization. Int. J. Geom. Meth. Math. Phys.
**2019**, 16, 1950017. [Google Scholar] [CrossRef] - Hatfield, B. Quantum Field Theory of Point Particles and Strings; Addison-Wesley: Reading, MA, USA, 1992. [Google Scholar]
- Jackiw, R. Analysis on infinite dimensional manifolds: Schrödinger representation for quantized fields. In Field Theory and Particle Physics; Éboli, O., Gomes, M., Santoro, A., Eds.; World Scientific: Singapore, 1990; pp. 78–143. [Google Scholar]
- Freese, K.; Hill, C.T.; Mueller, M.T. Covariant functional Schrödinger formalism and application to the Hawking effect. Nucl. Phys. B
**1985**, 255, 693–716. [Google Scholar] [CrossRef] - Pi, S.-Y. Quantum field theory in flat Robertson-Walker space-time: Functional Schrödinger picture. In Field Theory and Particle Physics; Éboli, O., Gomes, M., Santoro, A., Eds.; World Scientific: Singapore, 1990; pp. 144–195. [Google Scholar]
- Long, D.V.; Shore, G.M. The Schrödinger wave functional and vacuum states in curved spacetime. Nucl. Phys. B
**1998**, 530, 247–278. [Google Scholar] [CrossRef][Green Version] - Long, D.V.; Shore, G.M. The Schrödinger wave functional and vacuum states in curved spacetime II: Boundaries and foliations. Nucl. Phys. B
**1998**, 530, 279–303. [Google Scholar] [CrossRef][Green Version] - Corichi, A.; Quevedo, H. Schrödinger representation for a scalar field on curved spacetime. Phys. Rev. D
**2002**, 66, 085025. [Google Scholar] [CrossRef][Green Version] - Corichi, A.; Cortez, J.; Quevedo, H. Schrödinger and Fock representation for a field theory on curved spacetime. Ann. Phys.
**2004**, 113, 446–478. [Google Scholar] [CrossRef][Green Version] - Kiefer, C. Quantum gravitational corrections to the functional Schrödinger equation. Phys. Rev. D
**1991**, 44, 1067–1076. [Google Scholar] [CrossRef] - Pollock, M.D. On the Dirac equation in curved space-time. Acta Phys. Pol. B
**2010**, 41, 1827–1846. [Google Scholar] - Saunders, D.J. The Geometry of Jet Bundles; Cambridge University Press: Cambridge, UK, 1989. [Google Scholar]
- Olver, P.J. Applications of Lie Groups to Differential Equations; Springer-Verlag: New York, NY, USA, 1986. [Google Scholar]
- Bertlmann, R.A. Anomalies in Quantum Field Theory; Clarendon Press: Oxford, UK, 2000. [Google Scholar]
- Klauder, J.R. Ultralocal scalar field models. Commun. Math. Phys.
**1970**, 18, 307–318. [Google Scholar] [CrossRef] - Klauder, J.R. Beyond Conventional Quantization; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Volterra, V.; Hostinský, B. Opérations Infinitésimales Linéaires; Gauthier-Villars: Paris, France, 1938. [Google Scholar]
- Slavík, A. Product Integration, Its History and Applications; Matfyzpress: Prague, Czech Republic, 2007; Available online: http://www.karlin.mff.cuni.cz/~slavik/product/product_integration.pdf (accessed on 14 November 2019).
- Gill, R.D. Product-integration. In Encyclopedia of Biostatistics; Armitage, P., Colton, T., Eds.; John Wiley & Sons: New York, NY, USA, 2005. [Google Scholar] [CrossRef]

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Kanatchikov, I.V.
Precanonical Structure of the Schrödinger Wave Functional of a Quantum Scalar Field in Curved Space-Time. *Symmetry* **2019**, *11*, 1413.
https://doi.org/10.3390/sym11111413

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Kanatchikov IV.
Precanonical Structure of the Schrödinger Wave Functional of a Quantum Scalar Field in Curved Space-Time. *Symmetry*. 2019; 11(11):1413.
https://doi.org/10.3390/sym11111413

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Kanatchikov, Igor V.
2019. "Precanonical Structure of the Schrödinger Wave Functional of a Quantum Scalar Field in Curved Space-Time" *Symmetry* 11, no. 11: 1413.
https://doi.org/10.3390/sym11111413